A185995 -id:A185995 - cp's OEIS frontend

A185994 A generalized q-Catalan number for q=2.

1, 2, 6, 26, 150, 1114, 10614, 131002, 2128278, 46294426, 1368718518, 55647242106, 3137452915158, 246751601425242, 27181289502625014, 4205716133932054842, 915890632125187606038, 281117681559174501597466, 121733516122198763782243638

Offset: 0

Paul Barry, Feb 08 2011

Hankel transform is A185995.

A185994 := proc(n)
    option remember ;
    if n = 0 then
        1;
    else
        2*procname(n-1)+add(2^k*procname(k)*procname(n-1-k),k=0..n-2) ;
    end if;
end proc:
seq(A185994(n),n=0..30) ; # R. J. Mathar, Feb 03 2025

G.f.: 1/(1-2x/(1-x/(1-4x/(1-2x/(1-8x/(1-4x/(1-16x/(1-8x/(1-... (continued fraction).
G.f.: 1/(1-2x-2x^2/(1-5x-8x^2/(1-10x-32x^2/(1-(2^2+2^4)x-2^7x^2/(1-(2^3+2^5)x-2^9x^2/(1-.... (continued fraction).
a(n)=if(n=0,1,2*a(n-1)+sum{k=0..n-2, 2^k*a(k)*a(n-1-k)}).

A260638 Irregular table: list of symmetric n X n matrices made from 2-binomial coefficients, read by rows, where the k-th row of any n X n matrix is filled with binomial coefficients [k-1,k-1]..[k+n-2,k-1] (for q=2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 7, 1, 7, 35, 1, 1, 1, 1, 1, 3, 7, 15, 1, 7, 35, 155, 1, 15, 155, 1395, 1, 1, 1, 1, 1, 1, 3, 7, 15, 31, 1, 7, 35, 155, 651, 1, 15, 155, 1395, 11811, 1, 31, 651, 11811, 200787, 1, 1, 1, 1, 1, 1, 1, 3, 7, 15, 31, 63, 1, 7, 35, 155

Offset: 1

Arkadiusz Wesolowski, Nov 11 2015

The determinant of the n X n matrix is 2^((n/6)*(2*n^2 - 3*n + 1)), that is, A185995(n-1).

The permanent is in A260639.

			The irregular table starts:
1;
1, 1;
1, 3;
1, 1, 1;
1, 3, 7;
1, 7, 35;

G. C. Greubel, Table of n, a(n) for n = 1..819

Cf. A000225, A006095, A185995, A260639.

Flatten@Flatten@Table[Table[QBinomial[r + c, r, 2], {r, 0, n}, {c, 0, n}], {n, 0, 5}]

A335010 a(n) = Product_{k=1..n} (2^k - 1)^k.

Original entry on oeis.org

1, 9, 3087, 156279375, 4474145825060625, 279739266376566114746420625, 149066302192479440045478394157489352699375, 2665022756273454714861436593121414601862237540654765380859375

Offset: 1

Vaclav Kotesovec, May 19 2020

nonn

Cf. A002884, A005329, A048651, A185995, A203303, A335011.

Table[Product[(2^k - 1)^k, {k, 1, n}], {n, 1, 10}]

a(n) = prod(k=1, n, (2^k-1)^k); \\ Michel Marcus, May 19 2020

a(n) ~ c * 2^(n*(n+1)*(2*n+1)/6), where c = A335011 = Product_{k>=1} (1 - 1/2^k)^k.

cp's OEIS Frontend

A185994 A generalized q-Catalan number for q=2.

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A260638 Irregular table: list of symmetric n X n matrices made from 2-binomial coefficients, read by rows, where the k-th row of any n X n matrix is filled with binomial coefficients [k-1,k-1]..[k+n-2,k-1] (for q=2).

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A335010 a(n) = Product_{k=1..n} (2^k - 1)^k.

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