A103210 -id:A103210 - cp's OEIS frontend

A133306 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)5^i6^(n-i), a(0)=1.

1, 6, 66, 906, 13926, 229326, 3956106, 70572066, 1291183806, 24095736726, 456879955026, 8776867331706, 170459895028566, 3341423256586206, 66023812564384026, 1313634856606430226, 26295597219228901806, 529199848207277494566, 10701116421278640683106, 217317899302044152030826

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Sixth column of array A103209.

The Hankel transform of this sequence is 30^C(n+1,2). - Philippe Deléham, Oct 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..745

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-22*x+1))/(10*x))) // G. C. Greubel, Feb 10 2018

CoefficientList[Series[(1-x-Sqrt[x^2-22*x+1])/(10*x), {x,0,50}], x] (* G. C. Greubel, Feb 10 2018 *)

x='x+O('x^30); Vec((1-x-sqrt(x^2-22*x+1))/(10*x)) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-22*z+1))/(10*z).
a(n) = Sum_{k, 0<=k<=n} A088617(n,k)*5^k.
a(n) = Sum_{k, 0<=k<=n} A060693(n,k)*5^(n-k).
a(n) = Sum_{k, 0<=k<=n} C(n+k, 2*k) 5^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 5*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 11*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
G.f.: 1/(1 - 6*x/(1 - 5*x/(1 - 6*x/(1 - 5*x/(1 - 6*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017
a(n) ~ 3^(1/4) * (11 + 2*sqrt(30))^(n + 1/2) / (10^(3/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 29 2021

A133307 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)6^i7^(n-i), a(0)=1.

Original entry on oeis.org

1, 7, 91, 1477, 26845, 522739, 10663471, 224939113, 4866571801, 107393779423, 2407939176643, 54700070934061, 1256249370578293, 29119953189833611, 680401905145643863, 16008309928027493713, 378930780842531820721, 9017843351806985482423, 215634517504141993966891

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Seventh column of array A103209.

The Hankel transform of this sequence is 42^C(n+1,2). - Philippe Deléham, Oct 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..700

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-26*x+1))/(12*x))) // G. C. Greubel, Feb 10 2018

a := n -> hypergeom([-n, n+1], [2], -6);
seq(round(evalf(a(n),32)),n=0..16); # Peter Luschny, May 23 2014

CoefficientList[Series[(1-x-Sqrt[x^2-26*x+1])/(12*x), {x,0,50}], x] (* G. C. Greubel, Feb 10 2018 *)

x='x+O('x^30); Vec((1-x-sqrt(x^2-26*x+1))/(12*x)) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-26*z+1))/(12*z).
a(n) = Sum_{k=0..n} A088617(n,k)*6^k .
a(n) = Sum_{k=0..n} A060693(n,k)*6^(n-k).
a(n) = Sum_{k=0..n} C(n+k, 2k)6^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 6*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 13*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
a(n) = hypergeom([-n, n+1], [2], -6). # Peter Luschny, May 23 2014
G.f.: 1/(1 - 7*x/(1 - 6*x/(1 - 7*x/(1 - 6*x/(1 - 7*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017
a(n) ~ 42^(1/4) * (13 + 2*sqrt(42))^(n + 1/2) / (12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 29 2021

A133308 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)7^i8^(n-i), a(0)=1.

Original entry on oeis.org

1, 8, 120, 2248, 47160, 1059976, 24958200, 607693640, 15175702200, 386555020552, 10004252294520, 262321706465736, 6953918939056440, 186059575955360136, 5018045415643478520, 136276936332343342152, 3723442515218861494200, 102281105054908404972040

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Eighth column of array A103209.

The Hankel transform of this sequence is 56^C(n+1,2). - Philippe Deléham, Oct 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..675

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-30*x+1))/(14*x))) // G. C. Greubel, Feb 10 2018

a := n -> hypergeom([-n, n+1], [2], -7);
seq(round(evalf(a(n), 32)), n=0..15); # Peter Luschny, May 23 2014

CoefficientList[Series[(1-x-Sqrt[x^2-30*x+1])/(14*x), {x,0,50}], x] (* G. C. Greubel, Feb 10 2018 *)

x='x+O('x^30); Vec((1-x-sqrt(x^2-30*x+1))/(14*x)) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-30*z+1))/(14*z).
a(n) = Sum_{k, 0<=k<=n} A088617(n,k)*7^k.
a(n) = Sum_{k, 0<=k<=n} A060693(n,k)*7^(n-k).
a(n) = Sum_{k, 0<=k<=n} C(n+k, 2k)7^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 7*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
Conjecture: (n+1)*a(n) + 15*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014
a(n) = hypergeom([-n, n+1], [2], -7). - Peter Luschny, May 23 2014
G.f.: 1/(1 - 8*x/(1 - 7*x/(1 - 8*x/(1 - 7*x/(1 - 8*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017

A133309 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)8^i9^(n-i), a(0)=1.

Original entry on oeis.org

1, 9, 153, 3249, 77265, 1968633, 52546473, 1450365921, 41058670113, 1185580310121, 34783088255289, 1033907690362257, 31070005849929969, 942384250116160857, 28812102048874578249, 887007207177728561601, 27473495809057571051073, 855518113376312857290441

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Ninth column of array A103209.

The Hankel transform of this sequence is 72^C(n+1,2). - Philippe Deléham, Oct 29 2007

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

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!( (1-x-Sqrt(x^2-34*x+1))/16 )); // G. C. Greubel, Feb 10 2018

Rest@ CoefficientList[ Series[(1-x-Sqrt[x^2-34*x+1])/16, {x, 0, 18}], x] (* Robert G. Wilson v, Oct 19 2007 *)
Table[-((3 I LegendreP[n, -1, 2, 17])/(2 Sqrt[2])), {n, 0, 20}] (* Vaclav Kotesovec, Aug 13 2013 *)

my(x='x+O('x^30)); Vec((1-x-sqrt(x^2-34*x+1))/16) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-34*z+1))/16.
a(n) = Sum_{k=0..n} A088617(n,k)*8^k.
a(n) = Sum_{k=0..n} A060693(n,k)*8^(n-k).
a(n) = Sum_{k=0..n} C(n+k, 2k)8^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 8*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
a(n) ~ sqrt(144+102*sqrt(2))*(17+12*sqrt(2))^n/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013
Recurrence: (n+1)*a(n) = 17*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Aug 13 2013
G.f.: 1/(1 - 9*x/(1 - 8*x/(1 - 9*x/(1 - 8*x/(1 - 9*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017

More terms from Robert G. Wilson v, Oct 19 2007

A219536 G.f. satisfies A(x) = 1 + x(A(x)^2 + 2A(x)^3).

Original entry on oeis.org

1, 3, 24, 255, 3102, 40854, 566934, 8164263, 120864390, 1827982362, 28122626760, 438720097638, 6923868098820, 110346550539780, 1773394661610258, 28707809007278775, 467677404522668742, 7661583171651546786, 126137791939032756960, 2085923447593966281378

Offset: 0

Paul D. Hanna, Nov 21 2012

			G.f.: A(x) = 1 + 3*x + 24*x^2 + 255*x^3 + 3102*x^4 + 40854*x^5 +...
Related expansions:
A(x)^2 = 1 + 6*x + 57*x^2 + 654*x^3 + 8310*x^4 + 112560*x^5 +...
A(x)^3 = 1 + 9*x + 99*x^2 + 1224*x^3 + 16272*x^4 + 227187*x^5 +...
The g.f. satisfies A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where
G(x) = 1 + 3*x + 15*x^2 + 93*x^3 + 645*x^4 + 4791*x^5 +...+ A103210(n)*x^n +...

G. C. Greubel, Table of n, a(n) for n = 0..790
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Column k=2 of A336574.

Cf. A003168, A103210, A219534, A219535.

CoefficientList[1/x*InverseSeries[Series[4*x^2/(1-x-Sqrt[1-10*x+x^2]), {x, 0, 20}], x],x] (* Vaclav Kotesovec, Dec 28 2013 *)

/* Formula A(x) = 1 + x*(A(x)^2 + 2*A(x)^3): */
{a(n)=my(A=1);for(i=1,n,A=1+x*(A^2+2*A^3) +x*O(x^n));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

/* Formula using Series Reversion: */
{a(n)=my(A=1,G=(1-x-sqrt(1-10*x+x^2+x^3*O(x^n)))/(4*x));A=(1/x)*serreverse(x/G);polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(2*n+k+1, n)/(2*n+k+1)); \\ Seiichi Manyama, Jul 26 2020

a(n) = sum(k=0, n, 2^(n-k)*binomial(2*n+1, k)*binomial(3*n-k, n-k))/(2*n+1); \\ Seiichi Manyama, Jul 26 2020

Let G(x) = (1-x - sqrt(1 - 10*x + x^2)) / (4*x), then g.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion(x/G(x)),
(2) A(x) = G(x*A(x)) and G(x) = A(x/G(x)),
where G(x) is the g.f. of A103210.
Recurrence: 4*n*(2*n+1)*(19*n-26)*a(n) = (2717*n^3 - 6435*n^2 + 4342*n - 840)*a(n-1) + 2*(n-2)*(2*n-3)*(19*n-7)*a(n-2). - Vaclav Kotesovec, Dec 28 2013
a(n) ~ (3/19)^(1/4) * (5+sqrt(57)) * ((143 + 19*sqrt(57))/16)^n / (16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 28 2013
From Seiichi Manyama, Jul 26 2020: (Start)
a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(2*n+k+1,n)/(2*n+k+1).
a(n) = (1/(2*n+1)) * Sum_{k=0..n} 2^(n-k) * binomial(2*n+1,k) * binomial(3*n-k,n-k). (End)
From Seiichi Manyama, Aug 10 2023: (Start)
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 3^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 3^k * 2^(n-k) * binomial(n,k) * binomial(2*n,k-1) for n > 0. (End)
a(n) = (-1)^(n+1) * (3/n) * Jacobi_P(n-1, 1, n+1, -5) for n >= 1. - Peter Bala, Sep 08 2024

A152601 a(n) = Sum_{k=0..n} C(n+k,2k)A000108(k)3^k*2^(n-k).

Original entry on oeis.org

1, 5, 40, 395, 4360, 51530, 637840, 8163095, 107140360, 1434252230, 19507077040, 268796321870, 3744480010960, 52647783144980, 746145741252640, 10648007952942095, 152877753577617160, 2206713692628578030

Offset: 0

Paul Barry, Dec 09 2008

Hankel transform is 15^C(n+1,2).

Cf. A103211, A103210.
Cf. A088617, A060693, A090181, A131198, A133336, A086810
Cf. A152600.

a(n) = A152600(n+1)/2.
a(n) = Sum_{k=0..n} A088617(n,k)*3^k*2^(n-k) = Sum_{k=0..n} A060693(n,k)*2^k*3^(n-k). - Philippe Deléham, Dec 10 2008
a(n) = Sum_{k=0..n} A090181(n,k)*5^k*3^(n-k). - Philippe Deléham, Dec 10 2008
a(n) = Sum_{k=0..n} A131198(n,k)*3^k*5^(n-k). - Philippe Deléham, Dec 10 2008
a(n) = Sum_{k=0..n} A133336(n,k)*(-2)^k*5^(n-k) = Sum_{k=0..n} A086810(n,k)*5^k*(-2)^(n-k). - Philippe Deléham, Dec 10 2008
G.f.: 1/(1-5x/(1-3x/(1-5x/(1-3x/(1-5x/(1-3x/(1-5x/(1-... (continued fraction). - Philippe Deléham, Nov 28 2011
Conjecture: (n+1)*a(n) +8*(-2*n+1)*a(n-1) +4*(n-2)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
G.f.: 1/G(x), with G(x) = 1-2*x-(3*x)/G(x) (continued fraction). - Nikolaos Pantelidis, Jan 09 2023

A348912 G.f. A(x) satisfies A(x) = (1 + 2 * x * A(x)^3) / (1 - x).

Original entry on oeis.org

1, 3, 21, 201, 2217, 26535, 335001, 4391553, 59203137, 815580507, 11430639165, 162470033625, 2336381642649, 33930648153615, 496935405133617, 7331179445170689, 108846406625097729, 1625145134034548019, 24385673680861258533, 367546405595389076649, 5561980053932228243529

Offset: 0

Ilya Gutkovskiy, Nov 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..828

Cf. A001764, A103210, A153231, A346626.

nmax = 20; A[] = 0; Do[A[x] = (1 + 2 x A[x]^3)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + 2 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 20}]

a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(n+2*k+1, n)/(n+2*k+1)); \\ Seiichi Manyama, Jul 24 2023

a(0) = 1; a(n) = a(n-1) + 2 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) ~ sqrt(-50 + 30*sqrt(3) + (22 - 12*sqrt(3))*(2*(sqrt(3) - 1))^(1/3) + (2*(sqrt(3) - 1))^(2/3)*(-11 + 7*sqrt(3)))/(4*sqrt(3*Pi)*(-1 + sqrt(3))^(3/2) * n^(3/2) * (1 + (3*(-1 + sqrt(3))^(1/3))/2^(2/3) - 3/(2*(-1 + sqrt(3)))^(1/3))^n). - Vaclav Kotesovec, Nov 04 2021
a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(n+2*k+1,n) / (n+2*k+1). - Seiichi Manyama, Jul 24 2023

A364431 G.f. satisfies A(x) = 1 + xA(x)(1 + 2*A(x)^3).

Original entry on oeis.org

1, 3, 27, 351, 5319, 87885, 1535517, 27898101, 521740197, 9977087439, 194191054263, 3834392341779, 76619557946475, 1546479815079321, 31482877148802873, 645689728734541929, 13328555370318744777, 276704344407952939131, 5773556701375333682355

Offset: 0

Seiichi Manyama, Jul 24 2023

nonn

Cf. A103210, A348912.

Cf. A364430, A364432.

A364431 := proc(n)
    add(2^k* binomial(n,k) * binomial(n+3*k+1,n) / (n+3*k+1),k=0..n) ;
end proc:
seq(A364431(n),n=0..70); # R. J. Mathar, Jul 25 2023

a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(n+3*k+1, n)/(n+3*k+1));

a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(n+3*k+1,n) / (n+3*k+1).

D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) +(-458*n^3 +201*n^2 +401*n -216)*a(n-1) +3*(-1105*n^3 +6549*n^2 -11384*n +5796)*a(n-2) +18*(-262*n^3 +2877*n^2 -10295*n +12006)*a(n-3) +27*(n-4)*(31*n^2 -314*n +735)*a(n-4) -81*(10*n-51) *(n-4)*(n-5)*a(n-5) +243*(n-5)*(n-6) *(n-4)*a(n-6)=0. - R. J. Mathar, Jul 25 2023

A107702 Triangle related to guillotine partitions of a k-dimensional box by n hyperplanes.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 15, 22, 1, 1, 5, 28, 93, 90, 1, 1, 6, 45, 244, 645, 394, 1, 1, 7, 66, 505, 2380, 4791, 1806, 1, 1, 8, 91, 906, 6345, 24868, 37275, 8558, 1, 1, 9, 120, 1477, 13926, 85405, 272188, 299865, 41586, 1, 1, 10, 153, 2248, 26845, 229326, 1204245, 3080596, 2474025, 206098, 1

Offset: 0

Paul Barry, May 21 2005

Row sums are A107703. Transpose of square array A103209, read by antidiagonals.

			Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  6,   1;
  1, 4, 15,  22,    1;
  1, 5, 28,  93,   90,     1;
  1, 6, 45, 244,  645,   394,     1;
  1, 7, 66, 505, 2380,  4791,  1806,    1;
  1, 8, 91, 906, 6345, 24868, 37275, 8558, 1;
  ...

Seiichi Manyama, Rows n = 0..139, flattened
E. Ackerman, G. Barequet, R. Y. Pinter and D. Romik, The number of guillotine partitions in d dimensions, Inf. Proc. Lett 98 (4) (2006) 162-167.

Diagonals: A000012, A006318, A103210, A103211, A133305, A133306, A133307, A133308, A133309. - Philippe Deléham, Dec 10 2008

Cf. A000384, A103209.

T(n, k) = sum(j=0, k, (n-k)^j*binomial(k+j, 2*j)*binomial(2*j, j)/(j+1)); \\ Seiichi Manyama, Oct 02 2023

Number triangle T(n, k)=if(k<=n, sum{j=0..k, C(k+j, 2j)(n-k)^j*C(j)}, 0), C(n) given by A000108.

A297704 Triangle read by rows, T(n,k) = binomial(n, k)*hypergeom2F1(k - n, n + 1, k + 2, -2) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 3, 1, 15, 6, 1, 93, 39, 9, 1, 645, 276, 72, 12, 1, 4791, 2073, 576, 114, 15, 1, 37275, 16242, 4689, 1020, 165, 18, 1, 299865, 131295, 38889, 8979, 1635, 225, 21, 1, 2474025, 1087080, 327960, 78888, 15510, 2448, 294, 24, 1

Offset: 0

Peter Luschny, Jan 07 2018

			Triangle starts:
[0]     1
[1]     3,     1
[2]    15,     6,    1
[3]    93,    39,    9,    1
[4]   645,   276,   72,   12,   1
[5]  4791,  2073,  576,  114,  15,  1
[6] 37275, 16242, 4689, 1020, 165, 18, 1

Peter Luschny, row n for n = 0..44

T(n, 0) = A103210(n).

Row sums are A243626(n+1).

T[n_, k_] := Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -2];
Table[T[n, k], {n, 0, 6}, {k, 0, n}] // Flatten

cp's OEIS Frontend

A133306 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*5^i*6^(n-i), a(0)=1.

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A133307 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*6^i*7^(n-i), a(0)=1.

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A133308 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*7^i*8^(n-i), a(0)=1.

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A133309 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*8^i*9^(n-i), a(0)=1.

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A219536 G.f. satisfies A(x) = 1 + x*(A(x)^2 + 2*A(x)^3).

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A152601 a(n) = Sum_{k=0..n} C(n+k,2k)*A000108(k)*3^k*2^(n-k).

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A348912 G.f. A(x) satisfies A(x) = (1 + 2 * x * A(x)^3) / (1 - x).

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A133306 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)5^i6^(n-i), a(0)=1.

A133307 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)6^i7^(n-i), a(0)=1.

A133308 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)7^i8^(n-i), a(0)=1.

A133309 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)8^i9^(n-i), a(0)=1.

A219536 G.f. satisfies A(x) = 1 + x(A(x)^2 + 2A(x)^3).

A152601 a(n) = Sum_{k=0..n} C(n+k,2k)A000108(k)3^k*2^(n-k).

A364431 G.f. satisfies A(x) = 1 + xA(x)(1 + 2*A(x)^3).