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

0,2

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).

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

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 *)

Prog

(Magma) /* A000027 convolved with A001106 (excluding 0): */ A001106:=func<n | n*(7*n-5)/2>; [&+[(n-i+1)*A001106(i): i in [1..n]]: n in [1..36]]; // Bruno Berselli, Dec 07 2012
(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
(GAP) List([0..40], n-> (7*n+4)*Binomial(n+3, 3)/4); # G. C. Greubel, Aug 29 2019

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.

Sequence in context: A232613 A357736 A057813 * A370534 A263227 A144932

Adjacent sequences: A051737 A051738 A051739 * A051741 A051742 A051743

Keyword

nonn,easy

Author

Barry E. Williams, Dec 07 1999

Status

approved