A349310 -id:A349310 - cp's OEIS frontend

A346626 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^3) / (1 - x).

1, 2, 8, 44, 280, 1936, 14128, 107088, 834912, 6652608, 53934080, 443467136, 3689334272, 30997608960, 262651640064, 2241857334528, 19257951946240, 166362924583936, 1444351689281536, 12595885932259328, 110287974501355520, 969178569410404352, 8544982917273509888, 75565732555028701184

Offset: 0

Ilya Gutkovskiy, Jul 25 2021

nonn

Partial sums of A213282.

Seiichi Manyama, Table of n, a(n) for n = 0..500
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See p. 26.

Cf. A001764, A006318, A199475, A213282, A349310, A349311, A349312, A349313, A349314.

nmax = 23; A[] = 0; Do[A[x] = (1 + x A[x]^3)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 23; CoefficientList[Series[Sum[(Binomial[3 k, k]/(2 k + 1)) x^k/(1 - x)^(3 k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + 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, 23}]

G.f.: Sum_{k>=0} ( binomial(3*k,k) / (2*k + 1) ) * x^k / (1 - x)^(3*k+1).
a(0) = 1; a(n) = a(n-1) + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) ~ 2^(n - 1/2) / (sqrt(3*Pi*(2 - (2 - sqrt(2))^(1/3)/2^(2/3) - 1/(2*(2 - sqrt(2)))^(1/3))) * n^(3/2) * (2 - 3/(sqrt(2) - 1)^(1/3) + 3*(sqrt(2) - 1)^(1/3))^n). - Vaclav Kotesovec, Nov 04 2021
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 2^(n-k) * binomial(n,k) * binomial(2*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024

A349311 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^5) / (1 - x).

Original entry on oeis.org

1, 2, 12, 112, 1232, 14832, 189184, 2512064, 34358784, 480745984, 6848734464, 99003237376, 1448575666176, 21411827808256, 319255531155456, 4796005997940736, 72520546008219648, 1102912584949792768, 16859182461720526848, 258886644574700699648

Offset: 0

Ilya Gutkovskiy, Nov 14 2021

nonn

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

Cf. A002294, A006318, A346626, A346647, A349310, A349312, A349313, A349314.

nmax = 19; A[] = 0; Do[A[x] = (1 + x A[x]^5)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 4 k, 5 k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 19}]

a(n) = Sum_{k=0..n} binomial(n+4*k,5*k) * binomial(5*k,k) / (4*k+1).
a(n) = F([(1+n)/4, (2+n)/4, (3+n)/4, (4+n)/4, -n], [1/2, 3/4, 1, 5/4], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 4*r) / (2 * 5^(3/4) * sqrt(2*Pi) * (1-r)^(1/4) * n^(3/2) * r^(n + 1/4)), where r = 0.0600920016324256496641829206872407657377702010870270617... is the real root of the equation 4^4 * (1-r)^5 = 5^5 * r. - Vaclav Kotesovec, Nov 15 2021

A349312 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^6) / (1 - x).

Original entry on oeis.org

1, 2, 14, 158, 2106, 30762, 476406, 7683926, 127692530, 2171184146, 37592376734, 660522703886, 11747865153962, 211093333172282, 3826315983647366, 69880933123237958, 1284661783610775010, 23753502514840942882, 441458929706855144494, 8242097867816771820926

Offset: 0

Ilya Gutkovskiy, Nov 14 2021

nonn

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

Cf. A002295, A006318, A346626, A346648, A349310, A349311, A349313, A349314.

nmax = 19; A[] = 0; Do[A[x] = (1 + x A[x]^6)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 5 k, 6 k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 19}]

a(n) = Sum_{k=0..n} binomial(n+5*k,6*k) * binomial(6*k,k) / (5*k+1).
a(n) = F([(1+n)/5, (2+n)/5, (3+n)/5, (4+n)/5, 1+n/5, -n], [2/5, 3/5, 4/5, 1, 6/5], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 5*r) / (2^(6/5) * 3^(7/10) * sqrt(5*Pi) * (1-r)^(3/10) * n^(3/2) * r^(n + 1/5)), where r = 0.04941755525635041337247049893940451999923592381716... is the smallest real root of the equation 5^5 * (1-r)^6 = 6^6 * r. - Vaclav Kotesovec, Nov 15 2021

A364407 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^3).

Original entry on oeis.org

1, 2, -6, 42, -350, 3234, -31878, 328426, -3494142, 38093442, -423344966, 4778162922, -54621614814, 631114404258, -7358619459654, 86472788963370, -1023093071862526, 12177054520248834, -145700860758056838, 1751559565664348842, -21145576694586256734

Offset: 0

Seiichi Manyama, Jul 23 2023

sign

Cf. A364393, A364408, A364409.

Cf. A349310, A364400.

A364407 := proc(n)
    if n = 0 then
        1;
    else
        (-1)^(n-1)*add( binomial(n,k) * binomial(n+3*k-2,n-1),k=0..n)/n ;
    end if;
end proc:
seq(A364407(n),n=0..70); # R. J. Mathar, Jul 25 2023

nmax = 20; A[_] = 1;
Do[A[x_] = 1 + x*(1 + 1/A[x]^3) + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)

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

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A349310.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(n+3*k-2,n-1) for n > 0.
D-finite with recurrence 3*n*(52*n-187)*(3*n-1) *(3*n-2)*a(n) +(14392*n^4 -70190*n^3 +56951*n^2 +50237*n -49500)*a(n-1) +3*(-17252*n^4 +205959*n^3 -851664*n^2 +1432459*n -815652)*a(n-2) +18*(-472*n^4 +1294*n^3 +36359*n^2 -226731*n +361171)*a(n-3) -27*(n-5)*(404*n^3 -2235*n^2 -4058*n +26406)*a(n-4) -81*(n-5)*(n-6) *(8*n^2+358*n-1785)*a(n-5) +243*(n-5)*(n-6) *(n-7)*(4*n-31)*a(n-6)=0. - R. J. Mathar, Jul 25 2023

A349313 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^7) / (1 - x).

Original entry on oeis.org

1, 2, 16, 212, 3320, 57024, 1038928, 19718512, 385668448, 7718866880, 157326086656, 3254310606208, 68142850580480, 1441588339943168, 30765576147680000, 661561298256228096, 14319744815795062272, 311756656998135770112, 6822215641015820419072

Offset: 0

Ilya Gutkovskiy, Nov 14 2021

nonn

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

Cf. A002296, A006318, A346626, A346649, A349310, A349311, A349312, A349314.

nmax = 18; A[] = 0; Do[A[x] = (1 + x A[x]^7)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 6 k, 7 k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} binomial(n+6*k,7*k) * binomial(7*k,k) / (6*k+1).
a(n) = F([(1+n)/6, (2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, -n], [1/3, 1/2, 2/3, 5/6, 1, 7/6], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 6*r) / (2 * 7^(2/3) * sqrt(3*Pi) * (1-r)^(1/3) * n^(3/2) * r^(n + 1/6)), where r = 0.04196526794785323647696104132939153750367778616407409162... is the real root of the equation 6^6 * (1-r)^7 = 7^7 * r. - Vaclav Kotesovec, Nov 15 2021

A366363 G.f. satisfies A(x) = (1 + x/A(x))/(1 - x).

Original entry on oeis.org

1, 2, 0, 4, -8, 32, -112, 432, -1696, 6848, -28160, 117632, -497664, 2128128, -9183488, 39940864, -174897664, 770452480, -3411959808, 15181264896, -67833868288, 304256253952, -1369404661760, 6182858317824, -27995941060608, 127100310290432, -578433619525632

Offset: 0

Seiichi Manyama, Oct 08 2023

sign

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A346626, A349310, A349311, A349312, A349313, A349314, A366364, A366365, A366366.

Cf. A366356.

A366363[n_]:=(-1)^(n-1)Sum[Binomial[2k-1,k]Binomial[k-1,n-k]/(2k-1),{k,0,n}];
Array[A366363,30,0] (* Paolo Xausa, Oct 20 2023 *)

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

G.f.: A(x) = -2*x / (1-sqrt(1+4*x*(1-x))).

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

A366364 G.f. satisfies A(x) = (1 + x/A(x)^2)/(1 - x).

Original entry on oeis.org

1, 2, -2, 14, -70, 426, -2714, 18118, -124814, 881042, -6339058, 46318334, -342769750, 2563781690, -19350683018, 147197511222, -1127334112542, 8685458120226, -67270210217186, 523472089991662, -4090668558473318, 32088204418069450, -252576222775705466

Offset: 0

Seiichi Manyama, Oct 08 2023

sign

Cf. A346626, A349310, A349311, A349312, A349313, A349314, A366363, A366365, A366366.

Cf. A366357.

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

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

A213336 G.f. satisfies A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 8, 64, 568, 5440, 54888, 574848, 6190872, 68132224, 762874568, 8663106496, 99536424952, 1155012037824, 13516570396968, 159340702404352, 1890451582396632, 22555522916988672, 270466907608087944, 3257754635421506368, 39397587357527547320

Offset: 0

Paul D. Hanna, Jun 09 2012

nonn

			G.f.: A(x) = 1 + x + 8*x^2 + 64*x^3 + 568*x^4 + 5440*x^5 + 54888*x^6 +...
G.f.: A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4 is g.f. of A002293:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A213335, A002293; variants: A006319, A213282.

Partial sums give A349310. - Seiichi Manyama, Oct 03 2023

/* G.f. A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4: */
{a(n)=local(A, G=1+x); for(i=1, n, G=1+x*G^4+x*O(x^n)); A=subst(G, x, x/(1-x+x*O(x^n))^4); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* G.f. A(x) = F(x*A(x)^4) where F(x) = 1 + x/F(-x)^4: */
{a(n)=local(F=1+x+x*O(x^n),A=1); for(i=1, n+1, F=1+x/subst(F^4, x, -x+x*O(x^n))); A=(serreverse(x/F^4)/x)^(1/4);polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

G.f. satisfies: A(x) = F(x*A(x)^4) where F(x) = 1 + x/F(-x)^4 is the g.f. of A213335.
G.f. A(x) satisfies: A(1 - G(-x)) = G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) = Sum_{k=0..n} binomial(n+3*k-1,n-k) * binomial(4*k,k)/(3*k+1). - Seiichi Manyama, Oct 03 2023

A349314 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^8) / (1 - x).

Original entry on oeis.org

1, 2, 18, 274, 4930, 97346, 2039570, 44524818, 1001773058, 23065953794, 540886665618, 12872727013522, 310135678438978, 7549240857128258, 185381380643501970, 4586875745951650706, 114244031335228433922, 2862001783406012428802, 72067481493990612275474

Offset: 0

Ilya Gutkovskiy, Nov 14 2021

nonn

In general, for k > 1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial(k*j,j) / ((k-1)*j+1) ~ (1-r)^(1/(k-1) - 1/2) * sqrt(1 + (k-1)*r) / (sqrt(2*Pi*(k-1)) * k^(1/(k-1) + 1/2) * n^(3/2) * r^(n + 1/(k-1))), where r is the smallest real root of the equation (k-1)^(k-1) * (1-r)^k = k^k * r. - Vaclav Kotesovec, Nov 15 2021

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

Cf. A006318, A007556, A346626, A346650, A349310, A349311, A349312, A349313.

nmax = 18; A[] = 0; Do[A[x] = (1 + x A[x]^8)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 7 k, 8 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} binomial(n+7*k,8*k) * binomial(8*k,k) / (7*k+1).
a(n) = F([(1+n)/7, (2+n)/7, (3+n)/7, (4+n)/7, (5+n)/7, (6+n)/7, 1+n/7, -n], [2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 7*r) / (2^(17/7) * sqrt(7*Pi) * (1-r)^(5/14) * n^(3/2) * r^(n + 1/7)), where r = 0.036466941615119756839260438459647497790132092200414533994... is the smallest real root of the equation 7^7 * (1-r)^8 = 8^8 * r. - Vaclav Kotesovec, Nov 15 2021

A366365 G.f. satisfies A(x) = (1 + x/A(x)^3)/(1 - x).

Original entry on oeis.org

1, 2, -4, 32, -240, 2064, -18816, 179264, -1762816, 17758976, -182342400, 1901196288, -20075427840, 214246524928, -2307200135168, 25039992254464, -273603550461952, 3007387399258112, -33230774508716032, 368915340555517952, -4112806343370539008

Offset: 0

Seiichi Manyama, Oct 08 2023

sign

Cf. A346626, A349310, A349311, A349312, A349313, A349314, A366363, A366364, A366366.

Cf. A366358.

a(n) = (-1)^(n-1)*sum(k=0, n, binomial(4*k-1, k)*binomial(3*k-1, n-k)/(4*k-1));

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(4*k-1,k) * binomial(3*k-1,n-k)/(4*k-1).

cp's OEIS Frontend

A346626 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^3) / (1 - x).

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A349311 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^5) / (1 - x).

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A349312 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^6) / (1 - x).

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A364407 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^3).

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A349313 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^7) / (1 - x).

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A366363 G.f. satisfies A(x) = (1 + x/A(x))/(1 - x).

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A366364 G.f. satisfies A(x) = (1 + x/A(x)^2)/(1 - x).

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A213336 G.f. satisfies A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A349314 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^8) / (1 - x).

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