A271567 Convolution of nonzero triangular numbers (A000217) and nonzero tetradecagonal numbers (A051866).

1, 17, 87, 287, 742, 1638, 3234, 5874, 9999, 16159, 25025, 37401, 54236, 76636, 105876, 143412, 190893, 250173, 323323, 412643, 520674, 650210, 804310, 986310, 1199835, 1448811, 1737477, 2070397, 2452472, 2888952, 3385448, 3947944, 4582809, 5296809, 6097119

Offset: 0

Author

Ilya Gutkovskiy, Apr 12 2016

Keywords

• nonn
• easy

Comments

More generally, the ordinary generating function for the convolution of triangular numbers and k-gonal numbers is (1 + (k - 3)*x)/(1 - x)^6.

Links

• OEIS Wiki, Figurate numbers
• Eric Weisstein's World of Mathematics, Triangular Number
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Crossrefs

Cf. A000217, A051866.

Cf. similar sequences of the convolution of triangular numbers and k-gonal numbers: A005585 (k=4), A051836 (k=5), A034263 (k=6), A027800 (k=7), A051843 (k=8), A051877 (k=9), A051878 (k=10), A051879 (k=11), A051880 (k=12), A056118 (k=13), this sequence (k=14).

Programs

Magma

/* From definition: */ P:=func; /*, where P(n, k) is the n-th k-gonal number, */ [&+[P(n+1-i, 3)*P(i, 14): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Apr 18 2016

Magma

[(n+1)*(n+2)*(n+3)*(n+4)*(12*n+5)/120: n in [0..40]]; // Bruno Berselli, Apr 18 2016

Mathematica

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 17, 87, 287, 742, 1638}, 40]
Table[(n + 1) (n + 2) (n + 3) (n + 4) (12 n + 5)/120, {n, 0, 40}]

Formula

O.g.f.: (1 + 11*x)/(1 - x)^6.

E.g.f.: (120 + 1920*x + 3240*x^2 + 1520*x^3 + 245*x^4 + 12*x^5)*exp(x)/120.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

a(n) = (n + 1)*(n + 2)*(n + 3)*(n + 4)*(12*n + 5)/120.

Sum_{n>=0} 1/a(n) = 20*((15552*(6*log(2) + 3*log(3) + 2*sqrt(3)*log(2 - sqrt(3)) + (2 - sqrt(3))*Pi) - 29449)/531867) = 1.07654258697...

Extensions

Edited by Bruno Berselli, Apr 18 2016