A329753 Doubly square pyramidal numbers.

0, 1, 55, 1015, 9455, 56980, 255346, 924490, 2850730, 7757035, 19096385, 43312841, 91753025, 183453270, 349074740, 636310340, 1117143236, 1897397285, 3129084635, 5026125195, 7884086595, 12104671656, 18225763270, 26957923950, 39228339150, 56233289775, 79500340101, 110961532605

Offset: 0

Author

Ilya Gutkovskiy, Nov 20 2019

Keywords

• nonn
• easy

Links

• Eric Weisstein's World of Mathematics, Square Pyramidal Number
• Index to sequences related to pyramidal numbers
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

Cf. A000290, A000330, A000583, A140236, A329754, A329755, A329756, A329757.

Programs

Mathematica

A000330[n_] := n (n + 1) (2 n + 1)/6; a[n_] := A000330[A000330[n]]; Table[a[n], {n, 0, 27}]
Table[Sum[k^2, {k, 0, n (n + 1) (2 n + 1)/6}], {n, 0, 27}]
nmax = 27; CoefficientList[Series[x (1 + 45 x + 510 x^2 + 1660 x^3 + 1715 x^4 + 519 x^5 + 30 x^6)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 55, 1015, 9455, 56980, 255346, 924490, 2850730, 7757035}, 28]

Formula

G.f.: x*(1 + 45*x + 510*x^2 + 1660*x^3 + 1715*x^4 + 519*x^5 + 30*x^6)/(1 - x)^10.

a(n) = A000330(A000330(n)).

a(n) = Sum_{k=0..A000330(n)} A000290(k).

a(n) = n *(2*n+1) *(n+2) *(n+1) *(2*n^2-n+3) *(2*n^3+3*n^2+n+3) /648. - R. J. Mathar, Nov 28 2019