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A265939 Central terms of triangle A102363. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..27.

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).

EXAMPLE

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.

MATHEMATICA

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

PROG

(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), ", "))

CROSSREFS

Cf. A102363.

Sequence in context: A272832 A273489 A097113 * A012814 A039919 A322875

Adjacent sequences:  A265936 A265937 A265938 * A265940 A265941 A265942

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 19 2016

STATUS

approved

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Last modified August 15 00:52 EDT 2020. Contains 336484 sequences. (Running on oeis4.)