A100325 Antidiagonal sums of square array A100324, which lists the self-convolutions of SHIFT(A003169).

1, 2, 6, 25, 130, 774, 5009, 34231, 242988, 1773767, 13229272, 100362848, 772016385, 6007208105, 47198747457, 373929821070, 2983774582206, 23958802697161, 193448157014605, 1569625544848531, 12791865082236462

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f. A(x) = (1+G003169(x))/(1-x-x*G003169(x)), where G003169(x) is the g.f. of A003169.

a(n) ~ (sqrt(3056686 + 12607266/sqrt(17)) * ((71 + 17*sqrt(17))/16)^n) / (10201 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 31 2023

Mathematica

f[n_]:= f[n]= If[n<2, 1, If[n==2, 3, ((324*n^2 -708*n +360)*f[n-1] -(371*n^2 -1831*n +2250)*f[n-2] + (20*n^2 -130*n +210)*f[n-3])/(16*n*(2*n-1)) ]]; (* f = A003169 *)
A[n_, k_]:= A[n, k]= If[n==0, f[k], If[k==0, 1, Sum[f[k-j]*A[n-1, j], {j, 0, k}]]]; (* A = 100324 *)
a[n_]:= a[n]= Sum[A[n-k, k], {k, 0, n}]; (* a = A100325 *)
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jan 31 2023 *)

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); if(n==0, 1, for(i=1, n, A=1+x*A/(2-A)^2); sum(k=0, n, polcoeff(A^(n-k+1), k)))}
(SageMath)
@CachedFunction
def f(n): # f = A003169
    if (n<2): return 1
    elif (n==2): return 3
    else: return ((324*n^2-708*n+360)*f(n-1) - (371*n^2-1831*n+2250)*f(n-2) + (20*n^2-130*n+210)*f(n-3))/(16*n*(2*n-1))
@CachedFunction
def A(n, k): # A = 100324
    if (n==0): return f(k)
    elif (k==0): return 1
    else: return sum( f(k-j)*A(n-1, j) for j in range(k+1) )
def T(n, k): return A(n-k, k)
def A100325(n): return sum( T(n, k) for k in range(n+1) )
[A100325(n) for n in range(41)] # G. C. Greubel, Jan 31 2023

Crossrefs

Cf. A003169, A100324.

Sequence in context: A132290 A030933 A299044 * A353180 A352428 A352436

Adjacent sequences: A100322 A100323 A100324 * A100326 A100327 A100328

Keyword

nonn

Author

Paul D. Hanna, Nov 17 2004

Status

approved