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 A213575 Antidiagonal sums of the convolution array A213573. 6
 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 July 12 08:07 EDT 2024. Contains 374239 sequences. (Running on oeis4.)