A002293 -id:A002293 - cp's OEIS frontend

A381988 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)), where B(x) = 1 + xB(x)^4 is the g.f. of A002293.

1, 2, 15, 313, 10773, 510981, 30876463, 2267990159, 196204786025, 19539828320905, 2201822913234771, 276969947671828995, 38473403439454795837, 5849221857618942870029, 966078641687956464576119, 172251173569831561500070711, 32975613823747758363130520529, 6746227557293225645352382744593

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A381987, A381989.

Cf. A002293, A002294, A346647, A377526.

a(n) = n!*sum(k=0, n, (k+1)^(n-k)*binomial(5*k+1, k)/((5*k+1)*(n-k)!));

Let F(x) be the e.g.f. of A377526. F(x) = B(x*A(x)) = exp( 1/4 * Sum_{k>=1} binomial(4*k,k) * (x*A(x))^k/k ).

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * A002294(k)/(n-k)!.

A382016 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 3, 37, 901, 32141, 1502701, 86737645, 5952271977, 473117681881, 42731313784921, 4321503662185601, 483709266378568429, 59360036142346311685, 7924411424305558028757, 1143251381667547987358581, 177245340974472998607370321, 29386977237154379581209716657

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A161629, A161630, A382015.

Cf. A002293, A380513.

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

a(n) = n! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(n+3*k,k)/((n+3*k) * (n-k-1)!) for n > 0.

A382059 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^3), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 7, 127, 3733, 152161, 7939261, 505087843, 37920697753, 3281899787137, 321700411900441, 35227497466867531, 4262151791317099285, 564639582580738851265, 81290104199287214904037, 12637400195063381931755731, 2109868901338065949399370161, 376504852688521502050554789889

Offset: 0

Seiichi Manyama, Mar 13 2025

nonn

Cf. A161629, A382058.
Cf. A364938, A382033.
Cf. A002293, A377548, A382034.

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

Let F(x) be the e.g.f. of A377548. F(x) = log(A(x))/x = B(x*A(x))^3.
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(x))^3 ) ).
a(n) = 3 * n! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.

A153399 G.f.: A(x) = F(xG(x)^3) where F(x) = G(x/F(x)) = 1 + xF(x)^3 is the g.f. of A001764 and G(x) = F(xG(x)) = 1 + xG(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 6, 45, 371, 3225, 29007, 267239, 2506605, 23842644, 229369064, 2227345899, 21801617643, 214862158025, 2130226863222, 21231722675274, 212613977684254, 2138164077605865, 21585420400120710, 218677042735538547

Offset: 0

Paul D. Hanna, Jan 15 2009

nonn

			G.f.: A(x) = F(x*G(x)^3) = 1 + x + 6*x^2 + 45*x^3 + 371*x^4 +... where
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 +...
G(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 969*x^4 + 7084*x^5 +...
A(x)^2 = 1 + 2*x + 13*x^2 + 102*x^3 + 868*x^4 + 7732*x^5 +...
A(x)^3 = 1 + 3*x + 21*x^2 + 172*x^3 + 1509*x^4 + 13764*x^5 +...
G(x)^3*A(x)^3 = 1 + 6*x + 45*x^2 + 371*x^3 + 3225*x^4 + 29007*x^5 +...

Cf. A000108, A001764, A002293; A153398, A153291.

{a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+1,k)/(3*k+1)*binomial(4*(n-k)+3*k,n-k)*3*k/(4*(n-k)+3*k)))}

a(n) = Sum_{k=0..n} C(3*k+1,k)/(3*k+1) * C(4*n-k,n-k)*3*k/(4*n-k) for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*G(x)^3*A(x)^3 where G(x) is the g.f. of A002293.
G.f. satisfies: A(x/F(x)) = F(x*F(x)^2) where F(x) is the g.f. of A001764.
G.f. satisfies: A(x/H(x)^2) = F(x*H(x)) where H(x) = 1 + x*H(x)^2 is the g.f. of A000108 (Catalan) and F(x) is the g.f. of A001764.

A186186 Expansion of 1/(1-x/(1-x)*A(x/(1-x))) where A(x) is the g.f. of A002293.

Original entry on oeis.org

1, 1, 3, 12, 63, 403, 2919, 22833, 187799, 1599718, 13984383, 124717327, 1130144932, 10375309228, 96290993853, 901915801437, 8514822062757, 80939662475426, 774025387921462, 7441380898249458, 71879194326339456, 697253570563306939, 6789448668631285664, 66340474776507262638

Offset: 0

Vladimir Kruchinin, Feb 14 2011

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A002293.

a(n)={if(n<1, n==0, sum(m=1, n, sum(k=m, n, binomial(n-1,k-1)*m/(3*k-2*m)*binomial(4*k-3*m-1,k-m))))} \\ Andrew Howroyd, Apr 17 2021

a(n) = Sum_{m=1..n} Sum_{k=m..n} binomial(n-1,k-1)*m/(3*k-2*m)*binomial(4*k-3*m-1,k-m), n>0, a(0)=1.

Terms a(18) and beyond from Andrew Howroyd, Apr 17 2021

A380605 Expansion of e.g.f. exp(2xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 2, 16, 260, 6544, 224672, 9797824, 518778752, 32332764160, 2319086302208, 188178044545024, 17043816700333568, 1704575787500099584, 186577340672207974400, 22185432394552519868416, 2847773562263558405439488, 392481896442656581445287936, 57805399208817471918851883008

Offset: 0

Seiichi Manyama, Jan 28 2025

nonn

Cf. A002293, A251574, A380515, A380606.

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

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A380515.

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

A380606 Expansion of e.g.f. exp(3xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 3, 27, 459, 11817, 411183, 18090459, 963856071, 60351513777, 4344290172891, 353515902334299, 32093341598006307, 3215888732193019353, 352572962113533923271, 41981774097966848444763, 5395346708265250105968927, 744369113570455426540767201, 109733083289828610273889269939

Offset: 0

Seiichi Manyama, Jan 28 2025

nonn

Cf. A002293, A251574, A380515, A380605.

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

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A380515.

a(n) = 3 * n! * Sum_{k=0..n-1} 3^(n-k) * binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.

A380643 Expansion of e.g.f. exp(xG(3x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 19, 865, 63289, 6402421, 827951491, 130454402149, 24246255965905, 5193341198368489, 1259626725043888051, 341256073037890028041, 102138911537774675080969, 33470594059698797005874845, 11918817613356955871120346979, 4582850483720783516657005897741

Offset: 0

Seiichi Manyama, Jan 29 2025

nonn

Cf. A002293, A380637.

Cf. A080893, A380640.

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

a(n) = 3 * n! * Sum_{k=0..n-1} 3^k * binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.
E.g.f. A(x) satisfies x = log(A(x)) * (1 - 3*log(A(x)))^3.
a(n) = 3^(n-1)*U(1-n, 2-4*n, 1/3), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 29 2025
E.g.f.: exp( Series_Reversion( x*(1-3*x)^3 ) ). - Seiichi Manyama, Mar 16 2025

A381940 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 2, 7, 51, 440, 4170, 41921, 438972, 4736281, 52286520, 587774685, 6705201456, 77426676892, 903251324476, 10629495065550, 126032922655030, 1504194199010435, 18056321542477095, 217859030049153565, 2640609137351540510, 32137554969392230950, 392580762083089376630

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A025227, A381787, A381937.

Cf. A002293, A346647, A365184.

a(n) = sum(k=0, n, binomial(5*k+1, k)*binomial(k+1, n-k)/(5*k+1));

a(n) = Sum_{k=0..n} binomial(5*k+1,k) * binomial(k+1,n-k)/(5*k+1).

a(n) = A365184(n) + A365184(n-1).

A381941 G.f. A(x) satisfies A(x) = (1 + x)^2 * B(x*A(x)), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 3, 10, 71, 644, 6461, 68971, 768054, 8820281, 103694479, 1241799996, 15095075897, 185769856443, 2310006893997, 28978952155943, 366315306556482, 4661272734504606, 59659914501348239, 767539555514812321, 9920124234695256009, 128744011085858468131, 1677087982747514335025

Offset: 0

Seiichi Manyama, Mar 10 2025

nonn

Cf. A366694, A367640, A381938.

Cf. A002293.

a(n) = sum(k=0, n, binomial(5*k+1, k)*binomial(2*k+2, n-k)/(5*k+1));

a(n) = Sum_{k=0..n} binomial(5*k+1,k) * binomial(2*k+2,n-k)/(5*k+1).

cp's OEIS Frontend

A381988 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A382016 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A382059 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^3), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A153399 G.f.: A(x) = F(x*G(x)^3) where F(x) = G(x/F(x)) = 1 + x*F(x)^3 is the g.f. of A001764 and G(x) = F(x*G(x)) = 1 + x*G(x)^4 is the g.f. of A002293.

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A186186 Expansion of 1/(1-x/(1-x)*A(x/(1-x))) where A(x) is the g.f. of A002293.

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A380605 Expansion of e.g.f. exp(2*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A380606 Expansion of e.g.f. exp(3*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A380643 Expansion of e.g.f. exp(x*G(3*x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A381940 G.f. A(x) satisfies A(x) = (1 + x) * B(x*A(x)), where B(x) is the g.f. of A002293.

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A381941 G.f. A(x) satisfies A(x) = (1 + x)^2 * B(x*A(x)), where B(x) is the g.f. of A002293.

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A381988 E.g.f. A(x) satisfies A(x) = exp(x) * B(xA(x)), where B(x) = 1 + xB(x)^4 is the g.f. of A002293.

A382016 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

A382059 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^3), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

A153399 G.f.: A(x) = F(xG(x)^3) where F(x) = G(x/F(x)) = 1 + xF(x)^3 is the g.f. of A001764 and G(x) = F(xG(x)) = 1 + xG(x)^4 is the g.f. of A002293.

A380605 Expansion of e.g.f. exp(2xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

A380606 Expansion of e.g.f. exp(3xG(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

A380643 Expansion of e.g.f. exp(xG(3x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.