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 A156886 a(n) = Sum_{k=0..n} C(n,k)*C(3*n+k,k) 5
 1, 5, 43, 416, 4239, 44485, 475780, 5156548, 56437231, 622361423, 6904185523, 76964141600, 861408728964, 9673849095708, 108954068684616, 1230185577016156, 13920106205444335, 157814104889538739 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n)=[x^n](1+5x+9x^2+7x^3+2x^4)^n. The coefficients (1,5,9,7,2) are the 5th row of A029635. LINKS Robert Israel, Table of n, a(n) for n = 0..937 P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4. FORMULA From Peter Bala, Feb 11 2018: (Start) a(n) = Sum_{k = 0..n} (-1)^(n-k)*C(n,k)*C(3*n+k,n)*2^k. a(n) = Sum_{k = 0..n} C(n,k)*C(3*n,k)*2^(n-k), 12*n*(3*n-1)*(3*n-2)*(238*n^2 - 663*n + 457)*a(n) = 2*(150416*n^5 - 644640*n^4 + 1020351*n^3 - 734334*n^2 + 237007*n - 26880)*a(n-1) - (3*n-3)*(3*n-4)*(3*n-5)*(238*n^2 - 187*n + 32)*a(n-2). (End) a(n) = P_n(0,2*n,3) where P_n(a,b,x) is the n-th Jacobi polynomial with parameters a and b. - Robert Israel, Feb 11 2018 a(n) ~ sqrt(1/3 + 11/(12*sqrt(7))) * ((316 + 119*sqrt(7))/54)^n / sqrt(Pi*n). - Vaclav Kotesovec, Jan 09 2023 MAPLE A156886 := proc(n) add(binomial(n, k)*binomial(3*n+k, k), k = 0..n); end proc: seq(A156886(n), n = 0..20); # Peter Bala, Feb 11 2018 MATHEMATICA a[n_] := Sum[ Binomial[n, k] Binomial[3n + k, k], {k, 0, n}]; Array[a, 21, 0] (* Robert G. Wilson v, Feb 11 2018 *) CROSSREFS Cf. A001850, A114496, A029635, A156887. Sequence in context: A241707 A322246 A306080 * A112115 A350117 A239265 Adjacent sequences: A156883 A156884 A156885 * A156887 A156888 A156889 KEYWORD easy,nonn AUTHOR Paul Barry, Feb 17 2009 STATUS approved

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Last modified September 10 09:32 EDT 2024. Contains 375786 sequences. (Running on oeis4.)