A247613 a(n) = Sum_{k=0..8} binomial(16,k)*binomial(n,k).

1, 17, 153, 969, 4845, 20349, 74613, 245157, 735471, 2031535, 5189327, 12316239, 27322191, 57029103, 112740255, 212383935, 383358645, 666220005, 1119362365, 1824861005, 2895653673, 4484253081, 6793194849, 10087438257, 14708950035, 21093714291

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Formula

G.f.: (1 + 8*x + 36*x^2 + 120*x^3 + 330*x^4 + 792*x^5 + 1716*x^6 + 3432*x^7 + 6435*x^8) / (1-x)^9.

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).

a(n) = (20160 - 15076944*n + 40499716*n^2 - 42247940*n^3 + 23174515*n^4 - 7234136*n^5 + 1335334*n^6 - 134420*n^7 + 6435*n^8) / 20160.

Mathematica

Table[(20160 - 15076944 n + 40499716 n^2 - 42247940 n^3 + 23174515 n^4 - 7234136 n^5 + 1335334 n^6 - 134420 n^7 + 6435 n^8)/20160, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 8 x + 36 x^2 + 120 x^3 + 330 x^4 + 792 x^5 + 1716 x^6 + 3432 x^7 + 6435 x^8)/(1 - x)^9, {x, 0, 40}], x]
Table[Sum[Binomial[16, k]Binomial[n, k], {k, 0, 8}], {n, 0, 30}] (* or *) LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 17, 153, 969, 4845, 20349, 74613, 245157, 735471}, 40] (* Harvey P. Dale, Mar 25 2015 *)

Prog

(Magma) m:=8; [&+[Binomial(2*m, k)*Binomial(n, k): k in [0..m]]: n in [0..40]]; /* or */ [(20160-15076944*n+40499716*n^2-42247940*n^3 +23174515*n^4-7234136*n^5+1335334*n^6-134420*n^7 +6435*n^8)/20160: n in [0..40]];
(Sage) m=8; [sum((binomial(2*m, k)*binomial(n, k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 23 2014

Crossrefs

Cf. A005408, A056108, A247608 - A247612.

Sequence in context: A139617 A188353 A162637 * A010969 A022582 A164543

Adjacent sequences: A247610 A247611 A247612 * A247614 A247615 A247616

Keyword

nonn,easy

Author

Vincenzo Librandi, Sep 23 2014

Status

approved