[seqfan] Permutations avoiding a pair sum

[rhhardin at att.net]

Is there an obvious proof that the (Empirical) below can be omitted?
I don't see a mapping that would prove it.

I'll submit these after seeing if more terms can be computed.

(The even terms of the first are A007060, which has a formula.
Maybe somebody can intuit a generalization.)

%I A000001
%S A000001 0,2,8,48,240,1968,13824,140160,1263360,15298560,168422400,2373073920,
%T A000001 30865121280,496199854080,7445355724800,134510244986880,2287168006717440
%N A000001 Number of permutations of 1..n with no adjacent pair summing to n+1
%C A000001 (Empirical) If a(n,k) is the number of permutations of 1..n with no adjacent pair
         summing to n+k, then a(n,k)=a(n,k+1) for n+k even.
%K A000001 nonn
%O A000001 2,2

%I A000002
%S A000002 2,2,12,48,336,1968,17760,140160,1543680,15298560,199019520,2373073920,
%T A000002 35611269120,496199854080,8437755432960,134510244986880,2556188496691200
%N A000002 Number of permutations of 1..n with no adjacent pair summing to n+2
%C A000002 (Empirical) If a(n,k) is the number of permutations of 1..n with no adjacent pair
         summing to n+k, then a(n,k)=a(n,k+1) for n+k even.
%K A000002 nonn
%O A000002 2,1

%I A000003
%S A000003 2,6,12,72,336,2640,17760,175680,1543680,18385920,199019520,2771112960,
%T A000003 35611269120,567422392320,8437755432960,151385755852800,2556188496691200
%N A000003 Number of permutations of 1..n with no adjacent pair summing to n+3
%C A000003 (Empirical) If a(n,k) is the number of permutations of 1..n with no adjacent pair
         summing to n+k, then a(n,k)=a(n,k+1) for n+k even.
%K A000003 nonn
%O A000003 2,1

%I A000004
%S A000004 2,6,24,72,480,2640,23040,175680,1895040,18385920,235791360,2771112960,
%T A000004 41153495040,567422392320,9572600217600,151385755852800,2858960008396800
%N A000004 Number of permutations of 1..n with no adjacent pair summing to n+4
%C A000004 (Empirical) If a(n,k) is the number of permutations of 1..n with no adjacent pair
         summing to n+k, then a(n,k)=a(n,k+1) for n+k even.
%K A000004 nonn
%O A000004 2,1

%I A000005
%S A000005 2,6,24,120,480,3600,23040,221760,1895040,22176000,235791360,3242695680,
%T A000005 41153495040,649729382400,9572600217600,170530956288000,2858960008396800
%N A000005 Number of permutations of 1..n with no adjacent pair summing to n+5
%C A000005 (Empirical) If a(n,k) is the number of permutations of 1..n with no adjacent pair
         summing to n+k, then a(n,k)=a(n,k+1) for n+k even.
%K A000005 nonn
%O A000005 2,1

%I A000006
%S A000006 2,6,24,120,720,3600,30240,221760,2338560,22176000,280143360,3242695680,
%T A000006 47638886400,649729382400,10872058982400,170530956288000,
%U A000006 3200021920972800,56707673547571200
%N A000006 Number of permutations of 1..n with no adjacent pair summing to n+6
%C A000006 (Empirical) If a(n,k) is the number of permutations of 1..n with no adjacent pair
         summing to n+k, then a(n,k)=a(n,k+1) for n+k even.
%K A000006 nonn
%O A000006 2,1

%I A000007
%S A000007 2,6,24,120,720,5040,30240,282240,2338560,26853120,280143360,3802982400,
%T A000007 47638886400,745007155200,10872058982400,192275074252800,
%U A000007 3200021920972800
%N A000007 Number of permutations of 1..n with no adjacent pair summing to n+7
%C A000007 (Empirical) If a(n,k) is the number of permutations of 1..n with no adjacent pair
         summing to n+k, then a(n,k)=a(n,k+1) for n+k even.
%K A000007 nonn
%O A000007 2,1

%I A000008
%S A000008 2,6,24,120,720,5040,40320,282240,2903040,26853120,333849600,3802982400,
%T A000008 55244851200,745007155200,12362073292800,192275074252800,
%U A000008 3584572069478400
%N A000008 Number of permutations of 1..n with no adjacent pair summing to n+8
%C A000008 (Empirical) If a(n,k) is the number of permutations of 1..n with no adjacent pair
         summing to n+k, then a(n,k)=a(n,k+1) for n+k even.
%K A000008 nonn
%O A000008 2,1

%I A000009
%S A000009 2,6,24,120,720,5040,40320,362880,2903040,32659200,333849600,4470681600,
%T A000009 55244851200,855496857600,12362073292800,216999220838400,
%U A000009 3584572069478400
%N A000009 Number of permutations of 1..n with no adjacent pair summing to n+9
%C A000009 (Empirical) If a(n,k) is the number of permutations of 1..n with no adjacent pair
         summing to n+k, then a(n,k)=a(n,k+1) for n+k even.
%K A000009 nonn
%O A000009 2,1

%I A000010
%S A000010 2,6,24,120,720,5040,40320,362880,3628800,32659200,399168000,4470681600,
%T A000010 64186214400,855496857600,14073067008000,216999220838400,
%U A000010 4018570511155200
%N A000010 Number of permutations of 1..n with no adjacent pair summing to n+10
%C A000010 (Empirical) If a(n,k) is the number of permutations of 1..n with no adjacent pair
         summing to n+k, then a(n,k)=a(n,k+1) for n+k even.
%K A000010 nonn
%O A000010 2,1


--
rhhardin at mindspring.com
rhhardin at att.net (either)