A329757 Doubly octagonal pyramidal numbers.

0, 1, 765, 27435, 345415, 2469420, 12352956, 48294610, 157609530, 447989355, 1141711615, 2663460261, 5775482505, 11777133550, 22789550070, 42150245460, 74946834916, 128723876325, 214401953745, 347453633935, 549386792955, 849592039296, 1287617552320, 1915941609990, 2803320397950, 4038796372975

Offset: 0

Author

Ilya Gutkovskiy, Nov 20 2019

Keywords

• nonn
• easy

Links

• 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. A000567, A002414, A140236, A264892, A329753, A329754, A329755, A329756.

Programs

Mathematica

A002414[n_] := n (n + 1) (2 n - 1)/2; a[n_] := A002414[A002414[n]]; Table[a[n], {n, 0, 25}]
Table[Sum[k (3 k - 2), {k, 0, n (n + 1) (2 n - 1)/2}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[x (1 + 755 x + 19830 x^2 + 105370 x^3 + 158255 x^4 + 70629 x^5 + 7930 x^6 + 110 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 765, 27435, 345415, 2469420, 12352956, 48294610, 157609530, 447989355}, 26]

Formula

G.f.: x*(1 + 755*x + 19830*x^2 + 105370*x^3 + 158255*x^4 + 70629*x^5 + 7930*x^6 + 110*x^7)/(1 - x)^10.

a(n) = A002414(A002414(n)).

a(n) = Sum_{k=0..A002414(n)} A000567(k).

a(n) = n *(2*n-1) *(n+1) *(2*n^3+n^2-n+2) *(2*n^3+n^2-n-1) /8 . - R. J. Mathar, Nov 28 2019