

A125250


Square array, read by antidiagonals, where A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n < 1 or k < 1, otherwise A(n,k) = A(n2,k2) + A(n1,k2) + A(n2,k1) + A(n1,k1).


0



1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 5, 1, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 3, 11, 3, 0, 0, 0, 0, 0, 0, 1, 13, 13, 1, 0, 0, 0, 0, 0, 0, 0, 9, 26, 9, 0, 0, 0, 0, 0, 0, 0, 0, 4, 32, 32, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1, 26, 63, 26, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 80, 80, 14, 0, 0, 0, 0, 0
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OFFSET

1,13


COMMENTS

It appears that the main diagonal (1,1,2,5,11,...) is A051286 (Whitney number of level n of the lattice of the ideals of the fence of size 2 n) that the diagonals (0,1,2,5,13,...) adjacent to the main diagonal are A110320 (Number of blocks in all RNA secondary structures with n nodes) and that the nth antidiagonal sum = A094686(n1) (a Fibonacci convolution). The nth row sum = A002605(n).


LINKS

Table of n, a(n) for n=1..105.


FORMULA

A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n < 1 or k < 1, otherwise A(n,k) = A(n2,k2) + A(n1,k2) + A(n2,k1) + A(n1,k1).
From Peter Bala, Nov 07 2017: (Start)
T(n,k) = Sum_{i = floor((n+1)/2)..k} binomial(i,ni)* binomial(i,ki).
Square array = A026729 * transpose(A026729), where A026729 is viewed as a lower unit triangular array. Omitting the first row and column of square array = A030528 * transpose(A030528).
O.g.f. 1/(1  t*(1 + t)*x  t*(1 + t)*x^2) = 1 + (t + t^2)*x + (t + 2*t^2 + 2*t^3 + t^4)*x^2 + .... Cf. A109466 with o.g.f. 1/(1  t*x  t*x^2).
The nth row polynomial R(n,t) satisfies R(n,t) = R(n,1  t).
R(n,t) = (1)^n*sqrt(t*(1 + t))^n*U(n, 1/2*sqrt(t*(1 + t))), where U(n,x) denotes the nth Chebyshev polynomial of the second kind.
The sequence of row polynomials R(n,t) is a divisibility sequence of polynomials, that is, if m divides n then R(m,t) divides R(n,t) in the polynomial ring Z[t].
R(n,1) = A002605; R(n,2) = A057089. (End)


EXAMPLE

Array starts as:
1 0 0 0 0 0 0 ...
0 1 1 0 0 0 0 ...
0 1 2 2 1 0 0 ...
0 0 2 5 5 3 1 0 ...
0 0 1 5 11 13 9 4 1 0...
0 0 0 3 13 26 32 26 14 5 1 0 ...
0 0 0 1 9 32 63 80 71 45 20 6 1 0 ...
0 0 0 0 4 26 80 153 201 191 135 71 27 7 1 0 ...
...


MATHEMATICA

T[n_, k_] := Sum[Binomial[i, ni] Binomial[i, ki], {i, Floor[(n+1)/2], k}];
Table[T[nk, k], {n, 0, 13}, {k, 0, n}] // Flatten (* JeanFrançois Alcover, Nov 12 2019 *)


PROG

(PARI) A=matrix(22, 22); A[1, 1]=1; A[2, 2]=1; A[2, 1]=0; A[1, 2]=0; A[3, 2]=1; A[2, 3]=1; for(n=3, 22, for(k=3, 22, A[n, k]=A[n2, k2]+A[n1, k2]+A[n2, k1]+A[n1, k1])); for(n=1, 22, for(i=1, n, print1(A[ni+1, i], ", ")))


CROSSREFS

Cf. A051286, A110320, A002605, A026729, A030528, A057089, A109466.
Sequence in context: A178176 A093569 A073091 * A048113 A028961 A110177
Adjacent sequences: A125247 A125248 A125249 * A125251 A125252 A125253


KEYWORD

nonn,tabl,easy


AUTHOR

Gerald McGarvey, Jan 15 2007


STATUS

approved



