A265939 Central terms of triangle A102363.

1, 5, 21, 86, 349, 1410, 5682, 22860, 91869, 368906, 1480486, 5938740, 23813746, 95462996, 382594884, 1533053976, 6141910749, 24603000666, 98541647454, 394644228516, 1580344177254, 6327940829436, 25336229584764, 101436400902696, 406088663224434, 1625644557045060, 6507440174581692, 26048128051398920

Offset: 0

Author

Paul D. Hanna, Feb 19 2016

Keywords

• nonn

Comments

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

Examples

			Triangle A102363 begins:
  1;
  2, 3;
  4, 5, 7;
  8, 9, 12, 15;
  16, 17, 21, 27, 31;
  32, 33, 38, 48, 58, 63;
  64, 65, 71, 86, 106, 121, 127;
  128, 129, 136, 157, 192, 227, 248, 255;
  256, 257, 265, 293, 349, 419, 475, 503, 511, 512;
  ...
where the terms in this sequence form the central terms in the above triangle.
RELATED SERIES.
Let G(x) be the g.f. of triangle A102363 in flattened form:
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 +...
where G(x) can be written
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 +...
then the terms in this sequence form the coefficients of x^(2*n*(n+1)) in G(x) for n>=0.
Note that the coefficient of x^(n*(n+1)/2) in G(x) equals 2^n for n>=0.
		

Crossrefs

Cf. A102363.

Programs

Mathematica

Table[(3*4^n - Binomial[2*n, n])/2, {n, 0, 30}] (* Vaclav Kotesovec, Feb 21 2016 *)

PARI

{tr(n) = ceil( (sqrt(8*n+9)-1)/2 )}
{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)}
for(n=0,30, print1(a(n),", "))

PARI

{a(n) = polcoeff( (3 - sqrt(1-4*x +x*O(x^n))) / (2*(1-4*x)) ,n)}
for(n=0,30, print1(a(n),", "))

Python

from math import comb
def A265939(n): return (3<<(m:=n<<1))-comb(m,n)>>1 # Chai Wah Wu, Jun 07 2025

Formula

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

a(n) = (3*4^n - binomial(2*n, n))/2. - Vaclav Kotesovec, Feb 21 2016

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.g.f.: (3*exp(4*x) - exp(2*x)*BesselI(0,2*x))/2. - Stefano Spezia, Jun 07 2025