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 A265939 #22 Jun 08 2025 03:33:03
%S A265939 1,5,21,86,349,1410,5682,22860,91869,368906,1480486,5938740,23813746,
%T A265939 95462996,382594884,1533053976,6141910749,24603000666,98541647454,
%U A265939 394644228516,1580344177254,6327940829436,25336229584764,101436400902696,406088663224434,1625644557045060,6507440174581692,26048128051398920
%N A265939 Central terms of triangle A102363.
%C A265939 Triangle A102363 is constructed by a Pascal-like rule with left edge = 2^n, right edge = 2^(n+1)-1 (n>=0).
%F A265939 G.f.: (3 - sqrt(1-4*x)) / (2*(1-4*x)).
%F A265939 a(n) = (3*4^n - binomial(2*n, n))/2. - _Vaclav Kotesovec_, Feb 21 2016
%F A265939 a(n) = the coefficient of x^(2*n*(n+1)) in Sum_{n>=0} x^n * (1+x)^tr(n) = Sum_{n>=0} A102363(n)*x^n, where tr(n) = A002024(n+1) = floor(sqrt(2*n+1) + 1/2).
%F A265939 E.g.f.: (3*exp(4*x) - exp(2*x)*BesselI(0,2*x))/2. - _Stefano Spezia_, Jun 07 2025
%e A265939 Triangle A102363 begins:
%e A265939 1;
%e A265939 2, 3;
%e A265939 4, 5, 7;
%e A265939 8, 9, 12, 15;
%e A265939 16, 17, 21, 27, 31;
%e A265939 32, 33, 38, 48, 58, 63;
%e A265939 64, 65, 71, 86, 106, 121, 127;
%e A265939 128, 129, 136, 157, 192, 227, 248, 255;
%e A265939 256, 257, 265, 293, 349, 419, 475, 503, 511, 512;
%e A265939 ...
%e A265939 where the terms in this sequence form the central terms in the above triangle.
%e A265939 RELATED SERIES.
%e A265939 Let G(x) be the g.f. of triangle A102363 in flattened form:
%e A265939 G(x) = 1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 7*x^5 + 8*x^6 + 9*x^7 + 12*x^8 + 15*x^9 + 16*x^10 + 17*x^11 + 21*x^12 + 27*x^13 + 31*x^14 + 32*x^15 +...
%e A265939 where G(x) can be written
%e A265939 G(x) = (1+x) + x*(1+x)^2 + x^2*(1+x)^2 + x^3*(1+x)^3 + x^4*(1+x)^3 + x^5*(1+x)^3 + x^6*(1+x)^4 + x^7*(1+x)^4 + x^8*(1+x)^4 + x^9*(1+x)^4 + x^10*(1+x)^5 + x^11*(1+x)^5 + x^12*(1+x)^5 + x^13*(1+x)^5 + x^14*(1+x)^5 + x^15*(1+x)^6 +...
%e A265939 then the terms in this sequence form the coefficients of x^(2*n*(n+1)) in G(x) for n>=0.
%e A265939 Note that the coefficient of x^(n*(n+1)/2) in G(x) equals 2^n for n>=0.
%t A265939 Table[(3*4^n - Binomial[2*n, n])/2, {n, 0, 30}] (* _Vaclav Kotesovec_, Feb 21 2016 *)
%o A265939 (PARI) {tr(n) = ceil( (sqrt(8*n+9)-1)/2 )}
%o A265939 {a(n) = my(S, N=2*n*(n+1)); S = sum(m=0,N, x^m * (1+x +x*O(x^N))^tr(m) ); polcoeff(S, N)}
%o A265939 for(n=0,30, print1(a(n),", "))
%o A265939 (PARI) {a(n) = polcoeff( (3 - sqrt(1-4*x +x*O(x^n))) / (2*(1-4*x)) ,n)}
%o A265939 for(n=0,30, print1(a(n),", "))
%o A265939 (Python)
%o A265939 from math import comb
%o A265939 def A265939(n): return (3<<(m:=n<<1))-comb(m,n)>>1 # _Chai Wah Wu_, Jun 07 2025
%Y A265939 Cf. A102363.
%K A265939 nonn
%O A265939 0,2
%A A265939 _Paul D. Hanna_, Feb 19 2016