This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A271567 #27 Feb 16 2025 08:33:33
%S A271567 1,17,87,287,742,1638,3234,5874,9999,16159,25025,37401,54236,76636,
%T A271567 105876,143412,190893,250173,323323,412643,520674,650210,804310,
%U A271567 986310,1199835,1448811,1737477,2070397,2452472,2888952,3385448,3947944,4582809,5296809,6097119
%N A271567 Convolution of nonzero triangular numbers (A000217) and nonzero tetradecagonal numbers (A051866).
%C A271567 More generally, the ordinary generating function for the convolution of triangular numbers and k-gonal numbers is (1 + (k - 3)*x)/(1 - x)^6.
%H A271567 OEIS Wiki, <a href="http://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>
%H A271567 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>
%H A271567 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1)
%F A271567 O.g.f.: (1 + 11*x)/(1 - x)^6.
%F A271567 E.g.f.: (120 + 1920*x + 3240*x^2 + 1520*x^3 + 245*x^4 + 12*x^5)*exp(x)/120.
%F A271567 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).
%F A271567 a(n) = (n + 1)*(n + 2)*(n + 3)*(n + 4)*(12*n + 5)/120.
%F A271567 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...
%t A271567 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 17, 87, 287, 742, 1638}, 40]
%t A271567 Table[(n + 1) (n + 2) (n + 3) (n + 4) (12 n + 5)/120, {n, 0, 40}]
%o A271567 (Magma) /* From definition: */ P:=func<n, k | (n^2*(k-2)-n*(k-4))/2>; /*, 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
%o A271567 (Magma) [(n+1)*(n+2)*(n+3)*(n+4)*(12*n+5)/120: n in [0..40]]; // _Bruno Berselli_, Apr 18 2016
%Y A271567 Cf. A000217, A051866.
%Y A271567 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).
%K A271567 nonn,easy
%O A271567 0,2
%A A271567 _Ilya Gutkovskiy_, Apr 12 2016
%E A271567 Edited by _Bruno Berselli_, Apr 18 2016