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A213575 Antidiagonal sums of the convolution array A213573. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 6 12:35 EDT 2022. Contains 357264 sequences. (Running on oeis4.)