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 A156887 a(n) = Sum_{k=0..n} C(n,k)*C(4*n+k,k). 3
 1, 6, 64, 768, 9708, 126386, 1676956, 22548168, 306167324, 4188703512, 57649462164, 797294161824, 11071026740964, 154250752864812, 2155368246401224, 30192512693210888, 423859798484668188, 5961793387214958792, 83998039356129372448, 1185277027372535468544 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n)=[x^n] (1+6x+14x^2+16x^3+9x^4+2x^5)^n. The coefficients (1,6,14,16,9,2) are the 6th row of A029635. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..862 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 Conjecture: 576*n*(32901928701*n-65877527665)*(4*n-3)*(2*n-1)*(4*n-1)*a(n) +(-8795436181229177*n^5 +35251410418024655*n^4 -47934714902592853*n^3 +29414167990853161*n^2 -9060238526902314*n +1466702211905280)*a(n-1) +8*(10299715469615*n^5 -136961193094719*n^4 +872530072905392*n^3 -2699499511785411*n^2 +3902106377543903*n -2123717948975100)*a(n-2) -64*(2*n-5)*(4*n-9)*(n-2)*(27741827*n-2925269736)*(4*n-11)*a(n-3)=0. - R. J. Mathar, Feb 25 2015 From Peter Bala, Feb 11 2018: (Start) a(n) = Sum_{k = 0..n} (-1)^(n-k)*C(n,k)*C(4*n+k,n)*2^k. a(n) = Sum_{k = 0..n} C(n,k)*C(4*n,k)*2^(n-k). 8*(4*n)*(4*n-1)*(4*n-2)**(4*n-3)*(4633*n^3-19662*n^2+27593*n-12804)*a(n) = (137604733*n^7-859190528*n^6+2179882848*n^5-2890753162*n^4+2144669963*n^3-880916550*n^2+182941416*n-14515200)*a(n-1) - (4*n-4)*(4*n-5)*(4*n-6)*(4*n-7)*(4633*n^3-5763*n^2+2168*n-240)*a(n-2). A proof of Mathar's conjectured third-order recurrence above follows easily using this second-order recurrence.  (End) MAPLE A156887 := proc(n)     add(binomial(n, k)*binomial(4*n+k, k), k=0..n) ; end proc: # R. J. Mathar, Feb 25 2015 MATHEMATICA Table[Sum[Binomial[n, k]Binomial[4n+k, k], {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Jul 24 2018 *) PROG (PARI) {a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n+k, k))} \\ Seiichi Manyama, Feb 02 2019 CROSSREFS Cf. A001850, A114496, A156886. Sequence in context: A186668 A025609 A309186 * A239847 A264634 A296165 Adjacent sequences:  A156884 A156885 A156886 * A156888 A156889 A156890 KEYWORD easy,nonn AUTHOR Paul Barry, Feb 17 2009 STATUS approved

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Last modified June 21 06:55 EDT 2021. Contains 345358 sequences. (Running on oeis4.)