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

1,2

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).

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

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 *)

Prog

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

Crossrefs

Cf. A213564, A213500, A232603, A232604.

Sequence in context: A143895 A281767 A323799 * A319491 A034443 A304626

Adjacent sequences: A213572 A213573 A213574 * A213576 A213577 A213578

Keyword

nonn,easy

Author

Clark Kimberling, Jun 18 2012

Status

approved