A251574 -id:A251574 - cp's OEIS frontend

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

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.

A251660 Table of coefficients in functions R(n,x) defined by R(n,x) = exp( nxG(n,x)^(n-1) ) / G(n,x)^(n-1) where G(n,x) = 1 + x*G(n,x)^n, for rows n>=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 8, 1, 1, 1, 4, 21, 56, 1, 1, 1, 5, 40, 261, 592, 1, 1, 1, 6, 65, 712, 4833, 8512, 1, 1, 1, 7, 96, 1505, 18784, 120303, 155584, 1, 1, 1, 8, 133, 2736, 51505, 663424, 3778029, 3456896, 1, 1, 1, 9, 176, 4501, 115056, 2354725, 29480896, 143531433, 90501632, 1

Offset: 1

Paul D. Hanna, Dec 21 2014

			This table begins:
n=1: [1, 1,  1,   1,     1,       1,        1,           1, ...];
n=2: [1, 1,  2,   8,    56,     592,     8512,      155584, ...];
n=3: [1, 1,  3,  21,   261,    4833,   120303,     3778029, ...];
n=4: [1, 1,  4,  40,   712,   18784,   663424,    29480896, ...];
n=5: [1, 1,  5,  65,  1505,   51505,  2354725,   135258625, ...];
n=6: [1, 1,  6,  96,  2736,  115056,  6455376,   454666176, ...];
n=7: [1, 1,  7, 133,  4501,  224497, 14926387,  1245099709, ...];
n=8: [1, 1,  8, 176,  6896,  397888, 30584128,  2948178304, ...];
n=9: [1, 1,  9, 225, 10017,  656289, 57255849,  6262226721, ...];
n=10:[1, 1, 10, 280, 13960, 1023760, 99935200, 12226859200, ...]; ...
where e.g.f. of row n equals: exp( n*x*G(n,x)^(n-1) ) / G(n,x)^(n-1).
Related table of coefficients in G(n,x) = 1 + x*G(n,x)^n  begins:
n=1: [1, 1,  1,   1,    1,     1,      1,        1, ...];
n=2: [1, 1,  2,   5,   14,    42,    132,      429, ...];
n=3: [1, 1,  3,  12,   55,   273,   1428,     7752, ...];
n=4: [1, 1,  4,  22,  140,   969,   7084,    53820, ...];
n=5: [1, 1,  5,  35,  285,  2530,  23751,   231880, ...];
n=6: [1, 1,  6,  51,  506,  5481,  62832,   749398, ...];
n=7: [1, 1,  7,  70,  819, 10472, 141778 , 1997688, ...];
n=8: [1, 1,  8,  92, 1240, 18278, 285384,  4638348, ...];
n=9: [1, 1,  9, 117, 1785, 29799, 527085,  9706503, ...];
n=10:[1, 1, 10, 145, 2470, 46060, 910252, 18730855, ...]; ...

Cf. Rows: A243953, A251573, A251574, A251575, A251576, A251577, A251578, A251579, A251580.

{T(n,k)=local(G=1); for(i=0, k, G=1+x*G^n +x*O(x^k)); k!*polcoeff(exp(n*x*G^(n-1))/G^(n-1), k)}
/* Print as a rectangular table */
for(n=1, 10, for(k=0,10, print1(T(n,k), ", "));print(""))
/* Print as a flattened table */
for(n=0, 12, for(k=0,n, print1(T(n-k+1,k), ", "));)
/* Print the Related table of functions G(n,x) = 1 + x*G(n,x)^n */
{R(n,k)=local(G=1); for(i=0, k, G=1+x*G^n +x*O(x^k)); polcoeff(G, k)}
for(n=1, 10, for(k=0,10, print1(R(n,k), ", "));print(""))

/* Binomial sum formula for term T(n,k) */
{T(n,k) = if(k<=1,1,sum(j=0,k, n^j * k!/j! * binomial(n*(k-1)-j, k-j) * (j-1)/(k-1)))}
for(n=1, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))

E.g.f. of row n, R(n,x), for n>=1, satisfies:
(1) [x^k/k!] R(n,x)^(k+1) = n^(k-1) * (n+k) * (k+1)^(k-2) for k>=0.
(2) R(n,x) = exp( n*x*G(n,x)^(n-1) ) / G(n,x)^(n-1), where G(n,x) = 1 + x*G(n,x)^n.
(3) R'(n,x)/R(n,x) = G(n,x)^(n-1), where G(n,x) = 1 + x*G(n,x)^n.
T(n,k) = Sum_{j=0..k} n^j * k!/j! * binomial(n*(k-1)-j, k-j) * (j-1)/(k-1) for k>1, n>=1.

cp's OEIS Frontend

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

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A251660 Table of coefficients in functions R(n,x) defined by R(n,x) = exp( nxG(n,x)^(n-1) ) / G(n,x)^(n-1) where G(n,x) = 1 + x*G(n,x)^n, for rows n>=1.

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PARI

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cp's OEIS Frontend

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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Author

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PARI

Formula

A251660 Table of coefficients in functions R(n,x) defined by R(n,x) = exp( n*x*G(n,x)^(n-1) ) / G(n,x)^(n-1) where G(n,x) = 1 + x*G(n,x)^n, for rows n>=1.

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Programs

PARI

PARI

Formula

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.

A251660 Table of coefficients in functions R(n,x) defined by R(n,x) = exp( nxG(n,x)^(n-1) ) / G(n,x)^(n-1) where G(n,x) = 1 + x*G(n,x)^n, for rows n>=1.