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A247608 a(n) = Sum_{k=0..3} binomial(6,k)*binomial(n,k). 8
1, 7, 28, 84, 195, 381, 662, 1058, 1589, 2275, 3136, 4192, 5463, 6969, 8730, 10766, 13097, 15743, 18724, 22060, 25771, 29877, 34398, 39354, 44765, 50651, 57032, 63928, 71359, 79345, 87906, 97062, 106833, 117239, 128300, 140036, 152467, 165613, 179494 (list; graph; refs; listen; history; text; internal format)
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 (4,-6,4,-1).

FORMULA

G.f.: (1+3*x+6*x^2+10*x^3)/(1-x)^4.

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4).

a(n) = (6+31*n-15*n^2+20*n^3)/6.

a(n) = 1+6*Binomial(n,1)+15*Binomial(n,2)+20*Binomial(n,3).

MATHEMATICA

Table[(6 + 31 n - 15 n^2 + 20 n^3)/6, {n, 0, 50}] (* or *) CoefficientList[Series[(1 + 3 x + 6 x^2 + 10 x^3)/(1-x)^4, {x, 0, 50}], x]

PROG

(MAGMA) [(6+31*n-15*n^2+20*n^3)/6: n in [0..40]]; /* or */ [1+6*Binomial(n, 1)+15*Binomial(n, 2)+20*Binomial(n, 3): n in [0..40]]; /* or */ I:=[1, 7, 28, 84]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]

(PARI) Vec((1+3*x+6*x^2+10*x^3)/(1-x)^4 + O (x^50)) \\ Michel Marcus, Sep 22 2014

(Sage) m=3; [sum((binomial(2*m, k)*binomial(n, k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 22 2014

CROSSREFS

Cf. A005408, A056108.

Sequence in context: A163705 A162595 A073363 * A341136 A166322 A008499

Adjacent sequences:  A247605 A247606 A247607 * A247609 A247610 A247611

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Sep 22 2014

STATUS

approved

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Last modified May 13 10:31 EDT 2021. Contains 343839 sequences. (Running on oeis4.)