A247615 a(n) = Sum_{k=0..10} binomial(20,k)*binomial(n,k).

1, 21, 231, 1771, 10626, 53130, 230230, 888030, 3108105, 10015005, 30045015, 84504355, 223651350, 558350430, 1318250890, 2952624906, 6296642121, 12834146941, 25098124271, 47262174531, 85990654178, 151631858378, 259857912678, 433877085278, 707369215553

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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Formula

G.f.: (1 + 10*x + 55*x^2 + 220*x^3 + 715*x^4 + 2002*x^5 + 5005*x^6 + 11440*x^7 + 24310*x^8 + 48620*x^9 + 92378*x^10) / (1-x)^11.

a(n) = 11*a(n-1) -55*a(n-2) +165*a(n-3) -330*a(n-4) +462*a(n-5) -462*a(n-6)+330*a(n-7) -165*a(n-8) +55*a(n-9) -11*a(n-10) +a(n-11).

a(n) = 1 - 5512999*n/630 + 212329883*n^2/8400 - 134689309*n^3/4536 + 3453077689*n^4/181440 - 64212077*n^5/8640 + 80300707*n^6/43200 - 1817521*n^7/6048 + 1860157*n^8/60480 - 331721*n^9/181440 + 46189*n^10/907200.

Mathematica

CoefficientList[Series[(1 + 10 x + 55 x^2 + 220 x^3 + 715 x^4 + 2002 x^5 + 5005 x^6 + 11440 x^7 + 24310 x^8 + 48620 x^9 + 92378 x^10)/(1 - x)^11, {x, 0, 40}], x]
Table[Sum[Binomial[20, k]Binomial[n, k], {k, 0, 10}], {n, 0, 30}] (* or *) LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {1, 21, 231, 1771, 10626, 53130, 230230, 888030, 3108105, 10015005, 30045015}, 30] (* Harvey P. Dale, May 19 2015 *)

Prog

(Magma) m:=10; [&+[Binomial(2*m, k)*Binomial(n, k): k in [0..m]]: n in [0..40]]; /* or */ I:=[1, 21, 231, 1771, 10626, 53130, 230230, 888030, 3108105, 10015005, 30045015]; [n le 11 select I[n] else 11*Self(n-1) -55*Self(n-2) +165*Self(n-3) -330*Self(n-4) +462*Self(n-5) -462*Self(n-6) +330*Self(n-7) -165*Self(n-8) +55*Self(n-9) -11*Self(n-10) +Self(n-11): n in [1..40]];
(Sage) m=10; [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, A247609, A247610, A247611, A247612.

Sequence in context: A064322 A126902 A162646 * A010973 A022586 A321952

Adjacent sequences: A247612 A247613 A247614 * A247616 A247617 A247618

Keyword

nonn,easy

Author

Vincenzo Librandi, Sep 23 2014

Status

approved