A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

0, 1, 16, 70, 200, 455, 896, 1596, 2640, 4125, 6160, 8866, 12376, 16835, 22400, 29240, 37536, 47481, 59280, 73150, 89320, 108031, 129536, 154100, 182000, 213525, 248976, 288666, 332920, 382075, 436480, 496496, 562496, 634865, 714000, 800310, 894216, 996151

Offset: 0

Author

Bruno Berselli, Dec 08 2012

Keywords

• nonn
• easy

Comments

Partial sums of A172073.

Apart from 0, all terms are in A135021: a(n) = A135021(A034856(n+1)) with n>0.

Links

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

Crossrefs

Cf. A135021, A172073.

Cf. convolution of the natural numbers (A000027) with the k-gonal numbers (* means "except 0"):

k= 2 (A000027 ): A000292;

k= 3 (A000217 ): A000332 (after the third term);

k= 4 (A000290 ): A002415 (after the first term);

k= 5 (A000326 ): A001296;

k= 6 (A000384*): A002417;

k= 7 (A000566 ): A002418;

k= 8 (A000567*): A002419;

k= 9 (A001106*): A051740;

k=10 (A001107*): A051797;

k=11 (A051682*): A051798;

k=12 (A051624*): A051799;

k=13 (A051865*): A055268.

Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.

Programs

Magma

A051866:=func; [&+[(n-k+1)*A051866(k): k in [0..n]]: n in [0..37]];

Magma

I:=[0,1,16,70,200]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Mathematica

A051866[k_] := k (6 k - 5); Table[Sum[(n - k + 1) A051866[k], {k, 0, n}], {n, 0, 37}]
CoefficientList[Series[x (1 + 11 x) / (1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Formula

G.f.: x*(1+11*x)/(1-x)^5.

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

From Amiram Eldar, Feb 15 2022: (Start)

Sum_{n>=1} 1/a(n) = 3*(3*sqrt(3)*Pi + 27*log(3) - 17)/80.

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(6*sqrt(3)*Pi - 64*log(2) + 37)/80. (End)