A098658 -id:A098658 - cp's OEIS frontend

A322246 Expansion of g.f. 1/sqrt(1 - 10x - 11x^2).

1, 5, 43, 395, 3811, 37775, 381205, 3895925, 40193395, 417697775, 4366043473, 45852847265, 483447391309, 5114115365585, 54252753665083, 576948203182475, 6148667240501395, 65651351673108575, 702154850931542305, 7520927108084780225, 80666557496061224281, 866249916689104887005, 9312623039533986068863, 100216202771039576006495, 1079454220008183284872861

Offset: 0

Paul D. Hanna, Dec 09 2018

nonn

			G.f.: A(x) = 1 + 5*x + 43*x^2 + 395*x^3 + 3811*x^4 + 37775*x^5 + 381205*x^6 + 3895925*x^7 + 40193395*x^8 + 417697775*x^9 + 4366043473*x^10 + ...
such that A(x)^2 = 1/(1 - 10*x - 11*x^2).
RELATED SERIES.
exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + 5*x + 34*x^2 + 260*x^3 + 2137*x^4 + 18425*x^5 + 164395*x^6 + 1505075*x^7 + 14058979*x^8 + 133459055*x^9 + 1283753308*x^10 + ...

Robert Israel, Table of n, a(n) for n = 0..960

Cf. A098658, A322247.

List([0..30], n -> Sum([0..n], k-> (-1)^(n-k)*3^k*Binomial(n,k) *Binomial(2*k,k))); # G. C. Greubel, Dec 09 2018

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/Sqrt(1 - 10*x - 11*x^2) )); // G. C. Greubel, Dec 09 2018

f:= gfun:-rectoproc({{(11*n+11)*a(n)+(15+10*n)*a(n+1)+(-n-2)*a(n+2), a(0) = 1, a(1) = 5},a(n),remember):
map(f, [$0..40]); # Robert Israel, Dec 09 2018

CoefficientList[Series[1/Sqrt[1 - 10*x - 11*x^2], {x,0,30}], x] (* G. C. Greubel, Dec 09 2018 *)

/* Using generating function: */
{a(n) = polcoeff( 1/sqrt(1 - 10*x - 11*x^2 +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

/* Using binomial formula: */
{a(n) = sum(k=0,n, (-1)^(n-k)*3^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))

/* Using binomial formula: */
{a(n) = sum(k=0,n, 11^(n-k)*(-3)^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))

/* a(n) is central coefficient in (1 + 5*x + 9*x^2)^n */
{a(n) = polcoeff( (1 + 5*x + 9*x^2 +x*O(x^n))^n, n)}
for(n=0,30,print1(a(n),", "))

s=(1/sqrt(1 - 10*x - 11*x^2)).series(x, 30); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 09 2018

a(n) = Sum_{k=0..n} 11^(n-k) * (-3)^k * binomial(n,k)*binomial(2*k,k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * 3^k * binomial(n,k)*binomial(2*k,k).
a(n) equals the (central) coefficient of x^n in (1 + 5*x + 9*x^2)^n.
D-finite with recurrence: (11*n+11)*a(n)+(15+10*n)*a(n+1)+(-n-2)*a(n+2)=0. - Robert Israel, Dec 09 2018
a(n) ~ 11^(n + 1/2) / (2*sqrt(3*Pi*n)). - Vaclav Kotesovec, Dec 13 2018
E.g.f.: exp(5*x) * BesselI(0,6*x). - Ilya Gutkovskiy, Feb 02 2021
a(n) = 11^n*2F1([1/2, -n], [1], 12/11), where 2F1 is the hypergeometric function. - Stefano Spezia, Feb 02 2021
From Peter Bala, Oct 13 2024: (Start)
a(n) = Integral_{x = -1..11} x^n * w(x) dx, where w(x) = 1/( Pi*sqrt((1 + x)*(11 - x)) ) is positive on the interval (-1, 11). The weight function w(x) is singular at x = -1 and at x = 11 and is the solution of the Hausdorff moment problem.
Inverse binomial transform of A098658.
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r. (End)
a(n) = (1/4)^n * Sum_{k=0..n} (-1)^k * 11^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A101600 Expansion of g.f. c(3x)^2, where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 6, 45, 378, 3402, 32076, 312741, 3127410, 31899582, 330595668, 3471254514, 36848701764, 394807518900, 4263921204120, 46370143094805, 507343918566690, 5580783104233590, 61682339573108100, 684673969261499910, 7629224228913856140, 85308598196036755020

Offset: 0

Paul Barry, Dec 08 2004

Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016.

Cf. A050159, A101601, A101602, A098658.

Z[0]:=1: for k to 30 do Z[k]:=simplify(1/(1-3*z*Z[k-1])) od: g:=sum((Z[j]-Z[j-1]), j=1..30): gser:=series(g, z=0, 27): seq(coeff(gser, z, n)/3, n=1..19); # Zerinvary Lajos, May 21 2008

a[n_] := 3^n * CatalanNumber[n + 1]; Array[a, 20, 0] (* Amiram Eldar, May 15 2022 *)

G.f.: 4/(1+sqrt(1-12*x))^2.
a(n) = 3^n * A000108(n+1).
(n+2)*a(n) -6*(2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 15 2011
O.g.f. A(x) = 1/x*series reversion( x/(1 + 3*x)^2 ). 1 + x*A'(x)/A(x) = 1/sqrt(1 - 12*x) is the o.g.f. for A098658. - Peter Bala, Jul 17 2015
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 87/121 + 648*arcsin(1/(2*sqrt(3)))/(121*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 93/169 + 648*arcsinh(1/(2*sqrt(3)))/(169*sqrt(13)). (End)
E.g.f.: BesselI(1,6*z)*exp(6*z)/(3*z) where BesselI is the modified Bessel function of type I. - Karol A. Penson, Feb 17 2025

A119309 a(n) = binomial(2n,n) 6^n.

Original entry on oeis.org

1, 12, 216, 4320, 90720, 1959552, 43110144, 960740352, 21616657920, 489977579520, 11171488813056, 255928652808192, 5886359014588416, 135839054182809600, 3143703825373593600, 72933928748667371520

Offset: 0

Reinhard Zumkeller, May 14 2006

nonn

Number of lattice paths from (0,0) to (n,n) using three kinds of steps (1,0) and two kinds of steps (0,1). - Joerg Arndt, Jul 01 2011

Central terms of the triangles in A013620 and A038220.

			a(3) = binomial(2*3,3) * (6^3) = 20 * 216 = 4320. - _Indranil Ghosh_, Mar 03 2017

Indranil Ghosh, Table of n, a(n) for n = 0..400
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Cf. A000984, A013620, A038220, A059304, A098658.

Table[Binomial[2n,n]*(6^n), {n, 0, 15}] (* Indranil Ghosh, Mar 03 2017 *)

/* same as in A092566 but use */
steps=[[1,0], [1,0], [1,0], [0,1], [0,1]]; /* note repeated entries */
/* Joerg Arndt, Jun 30 2011 */

a(n)=binomial(2*n,n)*6^n \\ Charles R Greathouse IV, Mar 03 2017

import math
f=math.factorial
def C(n,r): return f(n)//f(r)//f(n-r)
def A119309(n): return C(2*n,n)*(6**n) # Indranil Ghosh, Mar 03 2017

a(n) = 6^n * A000984(n).
G.f.: 1/sqrt(1-24*x). - Zerinvary Lajos, Dec 20 2008 [Corrected by Joerg Arndt, Jul 01 2011]
D-finite with recurrence: n*a(n) +12*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 20 2020
a(n) = 2^n*A098658(n) = 3^n*A059304(n). - R. J. Mathar, Jan 20 2020
From Amiram Eldar, Jul 21 2020: (Start)
Sum_{n>=0} 1/a(n) = 24/23 + 24*sqrt(23)*arcsin(1/sqrt(24))/529.
Sum_{n>=0} (-1)^n/a(n) = 24/25 - 24*arcsinh(1/sqrt(24))/125. (End)
E.g.f.: exp(12*x) * BesselI(0,12*x). - Ilya Gutkovskiy, Sep 14 2021

A224881 Expansion of 1/(1 - 16*x)^(1/8).

Original entry on oeis.org

1, 2, 18, 204, 2550, 33660, 460020, 6440280, 91773990, 1325624300, 19354114780, 285033326760, 4227994346940, 63094684869720, 946420273045800, 14259398780556720, 215673406555920390, 3273161111260438860, 49824785804742235980, 760483572809223601800

Offset: 0

Paul D. Hanna, Jul 23 2013

nonn

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 204*x^3 + 2550*x^4 + 33660*x^5 + ...
where
A(x)^8 = 1 + 16*x + 256*x^2 + 4096*x^3 + 65536*x^4 + ... + 16^n*x^n + ...
Also,
A(x)^4 = 1 + 8*x + 96*x^2 + 1280*x^3 + 17920*x^4 + 258048*x^5 + ... + 4^n*A000984(n)*x^n + ...
A(x)^2 = 1 + 4*x + 40*x^2 + 480*x^3 + 6240*x^4 + 84864*x^5 + ... + 2^n*A004981(n)*x^n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..500

(1-b*x)^(-1/A003557(b)): A000984 (b=4), A004981 (b=8), A004987 (b=9), A098658 (b=12), this sequence (b=16), A034688 (b=25), A298799 (b=27), A004993 (b=36), A034835 (b=49).

Cf. A301271.

List([0..20],n->(2^n/Factorial(n))*Product([0..n-1],k->8*k+1)); # Muniru A Asiru, Jun 23 2018

seq(coeff(series(1/(1-16*x)^(1/8), x,50),x,n+1),n=0..20); # Muniru A Asiru, Jun 23 2018

CoefficientList[Series[1/(1-16*x)^(1/8), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 24 2013 *)

{a(n)=polcoeff(1/(1-16*x +x*O(x^n))^(1/8),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=(2^n/n!)*prod(k=0,n-1,8*k + 1)}
for(n=0,30,print1(a(n),", "))

a(n) = (2^n/n!) * Product_{k=0..n-1} (8*k + 1).

a(n) ~ 16^n/(GAMMA(1/8)*n^(7/8)). - Vaclav Kotesovec, Jul 24 2013

A298799 Expansion of (1-27*x)^(-1/9).

Original entry on oeis.org

1, 3, 45, 855, 17955, 398601, 9167823, 216098685, 5186368440, 126201632040, 3104560148184, 77049538223112, 1926238455577800, 48452305767226200, 1225151160114148200, 31118839466899364280, 793530406405933789140, 20305042752151835192700

Offset: 0

Seiichi Manyama, Jun 22 2018

nonn

Conjecture: a(p*n) == a(n) (mod p^2) for prime p == 1 (mod 9) and all positive integers n except those n of the form n = m*p + k for 0 <= m <= (p-1)/9 and 1 <= k <= (p-1)/9. Cf. A034171, A004981 and A004982. - Peter Bala, Dec 23 2019

Seiichi Manyama, Table of n, a(n) for n = 0..700

(1-b*x)^(-1/A003557(b)): A000984 (b=4), A004981 (b=8), A004987 (b=9), A098658 (b=12), A224881 (b=16), A034688 (b=25), this sequence (b=27), A004993 (b=36), A034835 (b=49).

Cf. A305991, A034171, A004981, A004982.

List([0..20],n->(3^n/Factorial(n))*Product([0..n-1],k->9*k+1)); # Muniru A Asiru, Jun 23 2018

seq(coeff(series((1-27*x)^(-1/9), x, n+1), x, n), n=0..20); # Muniru A Asiru, Jun 23 2018
# Alternative:
A298799 := n -> (-27)^n*binomial(-1/9, n):
seq(A298799(n), n=0..17); # Peter Luschny, Dec 26 2019

N=20; x='x+O('x^N); Vec((1-27*x)^(-1/9))

a(n) = 3^n/n! * Product_{k=0..n-1} (9*k + 1) for n > 0.
a(n) ~ 3^(3*n) / (Gamma(1/9) * n^(8/9)). - Vaclav Kotesovec, Jun 23 2018
From Peter Luschny, Dec 26 2019: (Start)
a(n) = (-27)^n*binomial(-1/9, n).
a(n) = n! * [x^n] hypergeom([1/9], [1], 27*x). (End)
D-finite with recurrence: n*a(n) +3*(-9*n+8)*a(n-1)=0. - R. J. Mathar, Jan 20 2020

A115903 Expansion of (1-12*x)^(-3/2).

Original entry on oeis.org

1, 18, 270, 3780, 51030, 673596, 8756748, 112586760, 1435481190, 18182761740, 229102797924, 2874198737592, 35927484219900, 447711726432600, 5564417171376600, 68998772925069840, 853859814947739270, 10547680067001485100, 130088054159684982900, 1602137088071909789400

Offset: 0

Paul Barry, Feb 02 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A002457, A098658.

[(3^n*Factorial(2*n)/Factorial(n)^2)*(2*n+1): n in [0..20]]; // Vincenzo Librandi, Jul 05 2011

CoefficientList[Series[(1-12x)^(-3/2),{x,0,20}],x] (* Harvey P. Dale, Oct 26 2016 *)

G.f.: 1/((1-12*x)*sqrt(1-12*x)).
a(n) = Jacobi_P(n,1/2,1/2,1)*12^n.
a(n) = 3^n*(2*n+1)*binomial(2*n,n) = 3^n*A002457(n).
a(n) = (2*n+1)*A098658(n).
D-finite with recurrence: n*a(n) - 6*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 07 2012
From Amiram Eldar, Jan 27 2024: (Start)
Sum_{n>=0} 1/a(n) = 12*arcsin(1/sqrt(12))/sqrt(11).
Sum_{n>=0} (-1)^n/a(n) = 12*arcsinh(1/sqrt(12))/sqrt(13). (End)

A360238 a(n) = [y^nx^n/n] log( Sum_{m>=0} (m + y)^(2m) * x^m ) for n >= 1.

Original entry on oeis.org

2, 42, 1376, 60934, 3377252, 224036904, 17282039280, 1519096411230, 149867251224092, 16398595767212452, 1971137737765484444, 258215735255164847944, 36617351885600586385222, 5588967440618883091216208, 913592455995572681826313856, 159241707066923571547572653630

Offset: 1

Paul D. Hanna, Feb 11 2023

nonn

Related sequence: A000984(n) = binomial(2*n,n) = [y^n*x^n/n] log( Sum_{m>=0} (1 + y)^(2*m) * x^m ) for n >= 1.

			L.g.f.: A(x) = 2*x + 42*x^2/2 + 1376*x^3/3 + 60934*x^4/4 + 3377252*x^5/5 + 224036904*x^6/6 + 17282039280*x^7/7 + 1519096411230*x^8/8 + ...
a(n) equals the coefficient of y^n*x^n/n in the logarithmic series:
log( Sum_{m>=0} (m + y)^(2*m) * x^m ) = (y^2 + 2*y + 1)*x + (y^4 + 12*y^3 + 42*y^2 + 60*y + 31)*x^2/2 + (y^6 + 30*y^5 + 297*y^4 + 1376*y^3 + 3348*y^2 + 4188*y + 2140)*x^3/3 + (y^8 + 56*y^7 + 1100*y^6 + 10792*y^5 + 60934*y^4 + 209464*y^3 + 436692*y^2 + 510952*y + 258779)*x^4/4 + (y^10 + 90*y^9 + 2945*y^8 + 49960*y^7 + 510160*y^6 + 3377252*y^5 + 14971780*y^4 + 44457000*y^3 + 85336175*y^2 + 96141170*y + 48446971)*x^5/5 + (y^12 + 132*y^11 + 6486*y^10 + 169236*y^9 + 2730921*y^8 + 29547696*y^7 + 224036904*y^6 + 1214958240*y^5 + 4717830978*y^4 + 12868488144*y^3 + 23497266672*y^2 + 25858665696*y + 12994749280)*x^6/6 + ...
Exponentiation yields the g.f. of A360239:
exp(A(x)) = 1 + 2*x + 23*x^2 + 502*x^3 + 16414*x^4 + 716936*x^5 + 39167817*x^6 + 2567058766*x^7 + 196159319943*x^8 + ... + A360239(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A360239, A360348, A266526, A000984, A059304, A098658.

{a(n) = n * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(2*m) *x^m ) +x*O(x^n) ), n, x), n, y)}
for(n=0,20,print1(a(n),", "))

a(n) ~ (1 - exp(-1)/4) * 2^(2*n) * n^(n + 1/2) / sqrt(Pi). - Vaclav Kotesovec, Feb 12 2023

cp's OEIS Frontend

A322246 Expansion of g.f. 1/sqrt(1 - 10*x - 11*x^2).

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A101600 Expansion of g.f. c(3x)^2, where c(x) is the g.f. of A000108.

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A119309 a(n) = binomial(2*n,n) * 6^n.

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A224881 Expansion of 1/(1 - 16*x)^(1/8).

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A298799 Expansion of (1-27*x)^(-1/9).

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A115903 Expansion of (1-12*x)^(-3/2).

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A360238 a(n) = [y^n*x^n/n] log( Sum_{m>=0} (m + y)^(2*m) * x^m ) for n >= 1.

A322246 Expansion of g.f. 1/sqrt(1 - 10x - 11x^2).

A119309 a(n) = binomial(2n,n) 6^n.

A360238 a(n) = [y^nx^n/n] log( Sum_{m>=0} (m + y)^(2m) * x^m ) for n >= 1.