A243953 -id:A243953 - cp's OEIS frontend

A369722 Expansion of e.g.f. exp( (3/2) * (1-sqrt(1-4*x)) ).

1, 3, 15, 117, 1305, 19323, 359559, 8084205, 213425361, 6475518675, 222088463199, 8497641269637, 358899729493545, 16585866328129803, 832523413971932055, 45105537151437499197, 2623613865509122341921, 163070009495928522691875

Offset: 0

Seiichi Manyama, Jan 30 2024

nonn

Cf. A001517, A101682, A243953, A369723, A369724.

Cf. A000108.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(3/2*(1-sqrt(1-4*x)))))

a(0) = 1; a(n) = Sum_{k=0..n-1} 3^(n-k) * (n-1+k)! / (k! * (n-1-k)!).

a(n) = 2*(2*n-3)*a(n-1) + 9*a(n-2).

A369723 Expansion of e.g.f. exp( 2 * (1-sqrt(1-4*x)) ).

Original entry on oeis.org

1, 4, 24, 208, 2464, 37824, 720256, 16450816, 439245312, 13440572416, 464007387136, 17847329869824, 757011972726784, 35108108023349248, 1767517592731090944, 96007679735852498944, 5596725706163142393856, 348533116657888468402176

Offset: 0

Seiichi Manyama, Jan 30 2024

nonn

Cf. A001517, A101682, A243953, A369722, A369724.

Cf. A000108.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*(1-sqrt(1-4*x)))))

a(0) = 1; a(n) = Sum_{k=0..n-1} 4^(n-k) * (n-1+k)! / (k! * (n-1-k)!).

a(n) = 2*(2*n-3)*a(n-1) + 16*a(n-2).

A369724 Expansion of e.g.f. exp( (5/2) * (1-sqrt(1-4*x)) ).

Original entry on oeis.org

1, 5, 35, 335, 4225, 67525, 1321075, 30751775, 832573025, 25745985125, 896177819875, 34698406783375, 1479737530398625, 68935386567921125, 3483762766656021875, 189846574063623209375, 11098195364856546690625, 692834276972696475053125

Offset: 0

Seiichi Manyama, Jan 30 2024

nonn

Cf. A001517, A101682, A243953, A369722, A369723.

Cf. A000108.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(5/2*(1-sqrt(1-4*x)))))

a(0) = 1; a(n) = Sum_{k=0..n-1} 5^(n-k) * (n-1+k)! / (k! * (n-1-k)!).

a(n) = 2*(2*n-3)*a(n-1) + 25*a(n-2).

A383796 Expansion of g.f.: exp(Sum_{n>=1} A295432(n)*x^n/n).

Original entry on oeis.org

1, 462, 396453, 425295010, 511915968714, 661059663660060, 895093835464198893, 1254056426977089876570, 1802794259810040618367902, 2644298823194748929633091780, 3941742074897786728895080586082, 5954164159064906497558129244865108, 9094122817144126105637193154022530612

Offset: 0

Karol A. Penson, Jun 11 2025

nonn

Cf. A295432, A255881, A243953, A166990, A229451.

seq(n)=Vec(exp(sum(n=1, n, (12*n)!*(3*n)!*(2*n)!*x^n/(n*((6*n)!)^2*(4*n)!*n!), O(x*x^n)))) \\ Andrew Howroyd, Jun 11 2025

G.f.: exp(Sum_{n>=1} (12*n)!*(3*n)!*(2*n)!*x^n/(n*((6*n)!)^2*(4*n)!*n!)).

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.

A384957 Expansion of g.f.: exp(Sum_{n>=1} A295433(n)*x^n/n).

Original entry on oeis.org

1, 990, 2206149, 6450139410, 21553605027306, 77957908218716988, 297118041166459732781, 1175248212459867447863562, 4779368947089383238327733950, 19858241947988743766121587718308, 83936671517628352407663509802203682, 359778601391313651280693986124971038388, 1560159110515342136997114532804454280500084

Offset: 0

Karol A. Penson, Jun 13 2025

nonn

Cf. A295432, A255881, A243953, A166990, A229451, A383796.

seq(n)=Vec(exp(sum(n=1, n, (12*n)!*n!*x^n/((8*n)!*(3*n)!*(2*n)!)/n, O(x*x^n)) )) \\ Andrew Howroyd, Jun 13 2025

G.f.: exp(Sum_{n>=1} (12*n)!*n!*x^n/((8*n)!*(3*n)!*(2*n)!)/n).

cp's OEIS Frontend

A369722 Expansion of e.g.f. exp( (3/2) * (1-sqrt(1-4*x)) ).

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A369723 Expansion of e.g.f. exp( 2 * (1-sqrt(1-4*x)) ).

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A369724 Expansion of e.g.f. exp( (5/2) * (1-sqrt(1-4*x)) ).

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A383796 Expansion of g.f.: exp(Sum_{n>=1} A295432(n)*x^n/n).

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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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PARI

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A384957 Expansion of g.f.: exp(Sum_{n>=1} A295433(n)*x^n/n).

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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.