A182960 -id:A182960 - cp's OEIS frontend

A182959 Expansion of o.g.f. 2(1+x)^2/(1-2x+sqrt(1-8*x)).

1, 5, 20, 96, 528, 3136, 19584, 126720, 841984, 5710848, 39376896, 275185664, 1944821760, 13875707904, 99807723520, 722997411840, 5269761884160, 38620004352000, 284405842575360, 2103530005463040, 15619068033761280

Offset: 0

Paul D. Hanna, Dec 31 2010

			G.f.: A(x) = 1 + 5*x + 20*x^2 + 96*x^3 + 528*x^4 + 3136*x^5 +...
where A(x*F(x)^3) = F(x) is the g.f. of A182960:
F(x) = 1 + 5*x + 95*x^2 + 2496*x^3 + 76063*x^4 + 2524161*x^5 +...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A182960, A001450, A167422, A372215.

CoefficientList[ Series[2 (1 + x)^2/(1 - 2 x + Sqrt[1 - 8 x]), {x, 0, 20}], x]  (* Robert G. Wilson v, Dec 31 2010 *)

{a(n)=polcoeff(2*(1+x)^2/(1-2*x+sqrt(1-8*x+x*O(x^n))),n)}

Let F(x) be the g.f. of A182960, then g.f. of this sequence satisfies:
* A(x) = F(x/A(x)^3) and A(x*F(x)^3) = F(x);
* A(x) = [x/Series_Reversion( x*F(x)^3 )]^(1/3).
G.f.: 1/2/x - 1/2 - x - (1+x)/x/G(0), where G(k)= 1 + 1/(1 - 4*x*(2*k+1)/(4*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
a(n) ~ 9*2^(3*n-2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013
From Peter Bala, Oct 04 2015: (Start)
O.g.f. A(x) = (1 + x)*(2*C(2*x) - 1), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108.
[x^n] A(x)^(3*n) = binomial(6*n,2*n). Cf. with the identity [x^n] ( (1 + x)*C(x) )^(5*n) = binomial(5*n,2*n) = A001450(n). (End)
Conjecture: D-finite with recurrence (n+1)*a(n) +(-7*n+3)*a(n-1) +4*(-2*n+5)*a(n-2)=0. - R. J. Mathar, Jan 22 2020
From Peter Bala, May 15 2023: (Start)
a(n) = 3*(2^n)*(3*n - 1)/(n*(n + 1)) * binomial(2*n-2,n-1) for n >= 2.
(n + 1)*(3*n - 4)*a(n) = 4*(2*n - 3)*(3*n - 1)*a(n-1) for n >= 3 with a(2) = 20. Mathar's conjectured second order recurrence above follows from this. (End)
[x^n] A(x)^n = A372215(n). - Peter Bala, Nov 07 2024

A381751 Expansion of exp( Sum_{k>=1} binomial(8k-1,2k-1) * x^k/k ).

Original entry on oeis.org

1, 7, 252, 12866, 767460, 50005591, 3449225652, 247579862356, 18301102679444, 1383742325041292, 106516121515030768, 8319491960857739258, 657680525420544788060, 52522142073165048614002, 4230907373618147894630904, 343379827862952363210331624, 28051180121294369965012932980

Offset: 0

Seiichi Manyama, Mar 06 2025

Cf. A006013, A079489, A182960, A381752, A381753, A381757, A381758.

Cf. A381745.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(8*k-1, 2*k-1)*x^k/k)))

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(8*k-1,2*k-1) * a(n-k).

A381752 Expansion of exp( Sum_{k>=1} binomial(10k-1,2k-1) * x^k/k ).

Original entry on oeis.org

1, 9, 525, 44067, 4338765, 467396050, 53346810991, 6339179481480, 775994115988525, 97182642466115275, 12392633418043399130, 1603634650155295053250, 210047857493659698690575, 27795006677556725604853840, 3710220786174094422360657000, 498998879378383167317202612400

Offset: 0

Seiichi Manyama, Mar 06 2025

Cf. A006013, A079489, A182960, A381751, A381753, A381757, A381758.

Cf. A381746.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(10*k-1, 2*k-1)*x^k/k)))

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(10*k-1,2*k-1) * a(n-k).

A381753 Expansion of exp( Sum_{k>=1} binomial(5k-1,2k-1) * x^k/k ).

Original entry on oeis.org

1, 4, 50, 846, 16495, 349240, 7803823, 181135830, 4324897697, 105543188190, 2620784850325, 66005699547352, 1682046970846570, 43291586055360034, 1123707191010320955, 29382536610737191930, 773229801368332554273, 20463493681189771623960

Offset: 0

Seiichi Manyama, Mar 06 2025

Cf. A006013, A079489, A182960, A381751, A381752, A381757, A381758.

Cf. A060941.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(5*k-1, 2*k-1)*x^k/k)))

a(n) = 2*sum(k=0, n, binomial(5*n+2*k+2, k)*binomial(5*n+2, n-k)/(5*n+2*k+2));

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(5*k-1,2*k-1) * a(n-k).
G.f.: B(x)^2, where B(x) is the g.f. of A060941.
a(n) = 2 * Sum_{k=0..n} binomial(5*n+2*k+2,k) * binomial(5*n+2,n-k)/(5*n+2*k+2).

A381757 Expansion of exp( Sum_{k>=1} binomial(7k-1,2k-1) * x^k/k ).

Original entry on oeis.org

1, 6, 161, 6062, 265868, 12720904, 643915209, 33905228350, 1838102210977, 101910583801012, 5751779249830131, 329359930638541776, 19087504000780665541, 1117418973753045781944, 65982722733895652916539, 3925378032146863676341770, 235048328495265879957413946

Offset: 0

Seiichi Manyama, Mar 06 2025

Cf. A006013, A079489, A182960, A381751, A381752, A381753, A381758.

Cf. A300386.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(7*k-1, 2*k-1)*x^k/k)))

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(7*k-1,2*k-1) * a(n-k).

G.f.: B(x)^2, where B(x) is the g.f. of A300386.

A381758 Expansion of exp( Sum_{k>=1} binomial(9k-1,2k-1) * x^k/k ).

Original entry on oeis.org

1, 8, 372, 24732, 1925394, 163883548, 14773987638, 1386341339430, 133994232166575, 13248555929274096, 1333732204895318366, 136243562694021684648, 14087033746990654649067, 1471456489458490198994856, 155042502964505871862313879, 16459391575059417875255359878

Offset: 0

Seiichi Manyama, Mar 06 2025

Cf. A006013, A079489, A182960, A381751, A381752, A381753, A381757.

Cf. A300387.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(9*k-1, 2*k-1)*x^k/k)))

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(9*k-1,2*k-1) * a(n-k).

G.f.: B(x)^2, where B(x) is the g.f. of A300387.

A381744 Expansion of exp( Sum_{k>=1} binomial(6k-1,2k) * x^k/k ).

Original entry on oeis.org

1, 10, 215, 5942, 186111, 6283192, 222992692, 8201608382, 309834609743, 11950890428170, 468707758663887, 18634632264615272, 749325132218313540, 30422303269317412048, 1245346665979469486376, 51343805279989437688334, 2130090659402456357279919, 88858984785475871013971710

Offset: 0

Seiichi Manyama, Mar 05 2025

Seiichi Manyama, Table of n, a(n) for n = 0..605

Cf. A079489, A381745, A381746.

Cf. A006013, A182960.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(6*k-1, 2*k)*x^k/k)))

G.f. A(x) satisfies A(x^2) = B(x) * B(-x), where B(x) is the g.f. of A006013.
a(n) = Sum_{k=0..2*n} (-1)^k * A006013(k) * A006013(2*n-k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(6*k-1,2*k) * a(n-k).
G.f.: B(x)^2, where B(x) is the g.f. of A182960.

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A182959 Expansion of o.g.f. 2*(1+x)^2/(1-2*x+sqrt(1-8*x)).

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A381751 Expansion of exp( Sum_{k>=1} binomial(8*k-1,2*k-1) * x^k/k ).

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A381752 Expansion of exp( Sum_{k>=1} binomial(10*k-1,2*k-1) * x^k/k ).

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A381753 Expansion of exp( Sum_{k>=1} binomial(5*k-1,2*k-1) * x^k/k ).

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A381757 Expansion of exp( Sum_{k>=1} binomial(7*k-1,2*k-1) * x^k/k ).

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A381758 Expansion of exp( Sum_{k>=1} binomial(9*k-1,2*k-1) * x^k/k ).

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A381744 Expansion of exp( Sum_{k>=1} binomial(6*k-1,2*k) * x^k/k ).

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A182959 Expansion of o.g.f. 2(1+x)^2/(1-2x+sqrt(1-8*x)).

A381751 Expansion of exp( Sum_{k>=1} binomial(8k-1,2k-1) * x^k/k ).

A381752 Expansion of exp( Sum_{k>=1} binomial(10k-1,2k-1) * x^k/k ).

A381753 Expansion of exp( Sum_{k>=1} binomial(5k-1,2k-1) * x^k/k ).

A381757 Expansion of exp( Sum_{k>=1} binomial(7k-1,2k-1) * x^k/k ).

A381758 Expansion of exp( Sum_{k>=1} binomial(9k-1,2k-1) * x^k/k ).

A381744 Expansion of exp( Sum_{k>=1} binomial(6k-1,2k) * x^k/k ).