A051740 Partial sums of A007584.

1, 11, 45, 125, 280, 546, 966, 1590, 2475, 3685, 5291, 7371, 10010, 13300, 17340, 22236, 28101, 35055, 43225, 52745, 63756, 76406, 90850, 107250, 125775, 146601, 169911, 195895, 224750, 256680, 291896, 330616, 373065, 419475, 470085, 525141

Offset: 0

Author

Barry E. Williams, Dec 07 1999

Keywords

• nonn
• easy

Comments

Convolution of A000027 with A001106 (excluding 0). - Bruno Berselli, Dec 07 2012

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
• Murray R.Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index to sequences related to pyramidal numbers
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A001106, A007584.

Cf. A093564 ((7, 1) Pascal, column m=4).

Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

Programs

GAP

List([0..40], n-> (7*n+4)*Binomial(n+3,3)/4); # G. C. Greubel, Aug 29 2019

Magma

/* A000027 convolved with A001106 (excluding 0): */ A001106:=func; [&+[(n-i+1)*A001106(i): i in [1..n]]: n in [1..36]]; // Bruno Berselli, Dec 07 2012

Maple

seq((7*n+4)*binomial(n+3,3)/4, n=0..40); # G. C. Greubel, Aug 29 2019

Mathematica

Table[(7*n+4)*Binomial[n+3,3]/4, {n,0,40}] (* G. C. Greubel, Aug 29 2019 *)
LinearRecurrence[{5,-10,10,-5,1},{1,11,45,125,280},40] (* Harvey P. Dale, May 18 2023 *)

PARI

vector(40, n, (7*n-3)*binomial(n+2,3)/4) \\ G. C. Greubel, Aug 29 2019

Sage

[(7*n+4)*binomial(n+3,3)/4 for n in (0..40)] # G. C. Greubel, Aug 29 2019

Formula

a(n) = binomial(n+3, 3)*(7*n+4)/4.

a(n) = (7*n+4)*binomial(n+3, 3)/4.

G.f.: (1+6*x)/(1-x)^5.

a(n) = A080852(7,n). - R. J. Mathar, Jul 28 2016

E.g.f.: (4! + 240*x + 288*x^2 + 88*x^3 + 7*x^4)*exp(x)/4!. - G. C. Greubel, Aug 29 2019