A213575 Antidiagonal sums of the convolution array A213573.

1, 10, 47, 158, 441, 1098, 2539, 5590, 11909, 24818, 50967, 103662, 209521, 421786, 846947, 1697990, 3400893, 6807618, 13622095, 27252190, 54513641, 109037930, 218088027, 436189878, 872395381, 1744808338, 3489636359

Offset: 1

Author

Clark Kimberling, Jun 18 2012

Keywords

• nonn
• easy

Links

• Clark Kimberling, Table of n, a(n) for n = 1..500
• S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
• Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Crossrefs

Cf. A213564, A213500, A232603, A232604.

Programs

GAP

List([1..30], n-> 13*2^(n+1)-(n^3+6*n^2+18*n+26)); # G. C. Greubel, Jul 25 2019

Magma

[13*2^(n+1)-(n^3+6*n^2+18*n+26): n in [1..30]]; // G. C. Greubel, Jul 25 2019

Mathematica

(* First program *)
b[n_]:= 2^(n-1); c[n_]:= n^2;
t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213573 *)
d = Table[t[n, n], {n, 1, 40}] (* A213574 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213575 *)
(* Additional programs *)
Table[Sum[k^3*2^(n-k),{k,0,n}],{n,1,30}] (* Vaclav Kotesovec, Nov 28 2013 *)

PARI

vector(30, n, 13*2^(n+1)-(n^3+6*n^2+18*n+26)) \\ G. C. Greubel, Jul 25 2019

Sage

[13*2^(n+1)-(n^3+6*n^2+18*n+26) for n in (1..30)] # G. C. Greubel, Jul 25 2019

Formula

a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 9*a(n-4) + 2*a(n-5).

G.f.: x*(1 + 4 x + x^2)/((1 - 2*x)*(1 - x)^4).

From Stanislav Sykora, Nov 27 2013: (Start)

a(n) = 2^n*Sum_{k=0..n} k^p*q^k, for p=3, q=1/2.

a(n) = 2^(n+1)*13 - (n^3 + 6*n^2 + 18*n + 26). (End)

a(n) = 2*a(n-1) + n^3. - Sochima Everton, Biereagu, Jul 14 2019

E.g.f.: 26*exp(2*x) - (26 +25*x +9*x^2 +x^3)*exp(x). - G. C. Greubel, Jul 25 2019