A265939 - OEIS

%I #13 Feb 21 2016 13:51:59

%S 1,5,21,86,349,1410,5682,22860,91869,368906,1480486,5938740,23813746,

%T 95462996,382594884,1533053976,6141910749,24603000666,98541647454,

%U 394644228516,1580344177254,6327940829436,25336229584764,101436400902696,406088663224434,1625644557045060,6507440174581692,26048128051398920

%N Central terms of triangle A102363.

%C Triangle A102363 is constructed by a Pascal-like rule with left edge = 2^n, right edge = 2^(n+1)-1 (n>=0).

%F G.f.: (3 - sqrt(1-4*x)) / (2*(1-4*x)).

%F a(n) = (3*4^n - binomial(2*n, n))/2. - _Vaclav Kotesovec_, Feb 21 2016

%F 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).

%e Triangle A102363 begins:

%e 1;

%e 2, 3;

%e 4, 5, 7;

%e 8, 9, 12, 15;

%e 16, 17, 21, 27, 31;

%e 32, 33, 38, 48, 58, 63;

%e 64, 65, 71, 86, 106, 121, 127;

%e 128, 129, 136, 157, 192, 227, 248, 255;

%e 256, 257, 265, 293, 349, 419, 475, 503, 511, 512; ...

%e where the terms in this sequence form the central terms in the above triangle.

%e RELATED SERIES.

%e Let G(x) be the g.f. of triangle A102363 in flattened form:

%e 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 where G(x) can be written

%e 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 then the terms in this sequence form the coefficients of x^(2*n*(n+1)) in G(x) for n>=0.

%e Note that the coefficient of x^(n*(n+1)/2) in G(x) equals 2^n for n>=0.

%t Table[(3*4^n - Binomial[2*n, n])/2, {n, 0, 30}] (* _Vaclav Kotesovec_, Feb 21 2016 *)

%o (PARI) {tr(n) = ceil( (sqrt(8*n+9)-1)/2 )}

%o {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 for(n=0,30, print1(a(n),", "))

%o (PARI) {a(n) = polcoeff( (3 - sqrt(1-4*x +x*O(x^n))) / (2*(1-4*x)) ,n)}

%o for(n=0,30, print1(a(n),", "))

%Y Cf. A102363.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 19 2016