A100325 - OEIS

%I #9 Jan 31 2023 05:02:43

%S 1,2,6,25,130,774,5009,34231,242988,1773767,13229272,100362848,

%T 772016385,6007208105,47198747457,373929821070,2983774582206,

%U 23958802697161,193448157014605,1569625544848531,12791865082236462

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

%H G. C. Greubel, <a href="/A100325/b100325.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%t 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 *)

%t 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 *)

%t a[n_]:= a[n]= Sum[A[n-k,k], {k,0,n}]; (* a = A100325 *)

%t Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, Jan 31 2023 *)

%o (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)))}

%o (SageMath)

%o @CachedFunction

%o def f(n): # f = A003169

%o     if (n<2): return 1

%o     elif (n==2): return 3

%o     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))

%o @CachedFunction

%o def A(n, k): # A = 100324

%o     if (n==0): return f(k)

%o     elif (k==0): return 1

%o     else: return sum( f(k-j)*A(n-1, j) for j in range(k+1) )

%o def T(n,k): return A(n-k, k)

%o def A100325(n): return sum( T(n,k) for k in range(n+1) )

%o [A100325(n) for n in range(41)] # _G. C. Greubel_, Jan 31 2023

%Y Cf. A003169, A100324.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 17 2004