cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A386230 G.f. A(x) satisfies A(x) = 1/( (1-x)^3 * (1 - x*A(x) - x^2*A'(x)) ).

Original entry on oeis.org

1, 4, 18, 114, 945, 9399, 106645, 1342028, 18409725, 272154510, 4300884555, 72225827628, 1283066570500, 24025524690426, 472822444534395, 9755834028122904, 210600429263424372, 4747647482075588598, 111583282733838959542, 2729989048854423409090, 69430953497076613542366
Offset: 0

Views

Author

Seiichi Manyama, Jul 16 2025

Keywords

Crossrefs

Cf. A088716, A321087, A386229.
Cf. A386212.

Programs

  • Mathematica
    terms = 21; A[] = 1; Do[A[x] = 1/((1-x)^3(1-x*A[x]-x^2*A'[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 16 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=binomial(i+2, 2)+(i+1)/2*sum(j=0, i-1, v[j+1]*v[i-j])); v;

Formula

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

A300620 Table of row functions R(n,x) that satisfy: [x^k] exp( k * R(n,x) ) = k^n * [x^(k-1)] exp( k * R(n,x) ) for k>=1, n>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 7, 30, 14, 1, 15, 207, 550, 85, 1, 31, 1290, 14226, 15375, 621, 1, 63, 7803, 340550, 1852800, 601398, 5236, 1, 127, 46830, 8086594, 215528250, 409408077, 31299268, 49680, 1, 255, 280647, 192663030, 25359510515, 280823532696, 142286748933, 2093655600, 521721, 1, 511, 1682130, 4605331346, 3013207159725, 197431364485587, 676005054191880, 73448832515952, 175312873125, 5994155
Offset: 1

Views

Author

Paul D. Hanna, Mar 12 2018

Keywords

Examples

			This table of coefficients T(n,k) begins:
n=1: [1, 1, 3, 14, 85, 621, 5236, ...];
n=2: [1, 3, 30, 550, 15375, 601398, 31299268, ...];
n=3: [1, 7, 207, 14226, 1852800, 409408077, 142286748933, ...];
n=4: [1, 15, 1290, 340550, 215528250, 280823532696, 676005054191880, ...];
n=5: [1, 31, 7803, 8086594, 25359510515, 197431364485587, ...];
n=6: [1, 63, 46830, 192663030, 3013207159725, 140620832995924134, ...];
n=7: [1, 127, 280647, 4605331346, 359881205186350, 100749338488125315273, 82972785219971584775198767, ...]; ...
such that row functions R(n,x) = Sum_{k>=1} T(n,k)*x^k satisfy:
[x^k] exp( k * R(n,x) ) = k^n * [x^(k-1)] exp( k * R(n,x) ) for k>=1.
Row functions R(n,x) begin:
R(1,x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 621*x^6 + 5236*x^7 + 49680*x^8 + ...
R(2,x) = x + 3*x^2 + 30*x^3 + 550*x^4 + 15375*x^5 + 601398*x^6 + 31299268*x^7 + ...
R(3,x) = x + 7*x^2 + 207*x^3 + 14226*x^4 + 1852800*x^5 + 409408077*x^6 + ...
R(4,x) = x + 15*x^2 + 1290*x^3 + 340550*x^4 + 215528250*x^5 + 280823532696*x^6 + ...
etc.

Links

Crossrefs

Cf. A088716 (row 1), A300617 (row 2), A300619 (row 3).

Programs

  • PARI
    {T(n,k) = my(A=[1]); for(i=1, k+1, A=concat(A, 0); V=Vec(Ser(A)^(#A-1)); A[#A] = ((#A-1)^n*V[#A-1] - V[#A])/(#A-1) ); polcoeff( log(Ser(A)), k)}
for(n=1, 8, for(k=1,8, print1(T(n,k), ", "));print(""))

A300625 Table of row functions R(n,x) that satisfy: [x^k] exp( k^n * R(n,x) ) = k^n * [x^(k-1)] exp( k^n * R(n,x) ) for k>=1, n>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 4, 27, 14, 1, 8, 243, 736, 85, 1, 16, 2187, 40448, 30525, 621, 1, 32, 19683, 2351104, 12519125, 1715454, 5236, 1, 64, 177147, 142475264, 6153518125, 6111917748, 123198985, 49680, 1, 128, 1594323, 8856272896, 3436799053125, 31779658925496, 4308276119854, 10931897664, 521721, 1, 256, 14348907, 558312194048, 2049047412828125, 212148041589128016, 287364845865893467, 4151360558858752, 1172808994833, 5994155
Offset: 1

Views

Author

Paul D. Hanna, Mar 12 2018

Keywords

Examples

			This table of coefficients T(n,k) begins:
n=1: [1, 1, 3, 14, 85, 621, 5236, 49680, ...];
n=2: [1, 2, 27, 736, 30525, 1715454, 123198985, 10931897664, ...];
n=3: [1, 4, 243, 40448, 12519125, 6111917748, 4308276119854, ..];
n=4: [1, 8, 2187, 2351104, 6153518125, 31779658925496, ...];
n=5: [1, 16, 19683, 142475264, 3436799053125, 212148041589128016, ...];
n=6: [1, 32, 177147, 8856272896, 2049047412828125, 1569837215111038900704, ...];
n=7: [1, 64, 1594323, 558312194048, 1256793474918203125, 12020665333382306853887808, ...]; ...
such that row functions R(n,x) = Sum_{k>=1} T(n,k)*x^k satisfy:
[x^k] exp( k^n * R(n,x) ) = k^n * [x^(k-1)] exp( k^n * R(n,x) ) for k>=1.
Row functions R(n,x) begin:
R(1,x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 621*x^6 + 5236*x^7 + 49680*x^8 + ...
R(2,x) = x + 2*x^2 + 27*x^3 + 736*x^4 + 30525*x^5 + 1715454*x^6 + ...
R(3,x) = x + 4*x^2 + 243*x^3 + 40448*x^4 + 12519125*x^5 + 6111917748*x^6 + ...
R(4,x) = x + 8*x^2 + 2187*x^3 + 2351104*x^4 + 6153518125*x^5 + ...
etc.

Links

Crossrefs

Cf. A088716 (row 1), A300591 (row 2), A300595 (row 3), A300597 (row 4).

Programs

  • PARI
    {T(n, k) = my(A=[1]); for(i=1, k+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^n)); A[#A] = ((#A-1)^n * V[#A-1] - V[#A])/(#A-1)^n ); polcoeff( log(Ser(A)), k)}
/* Print as a table of row functions: */
for(n=1, 8, for(k=1, 8, print1(T(n, k), ", ")); print(""))
/* Print as a flattened triangle: */
for(n=1, 12, for(k=1, n-1, print1(T(n-k, k), ", ")); )

A305118 a(n) = Sum_{k=0..n-1} ( 1 + a(k) * a(n-k-1) ) for n >= 1, a(0) = 1.

Original entry on oeis.org

1, 2, 6, 19, 66, 249, 996, 4148, 17784, 77939, 347516, 1571304, 7187288, 33196887, 154611392, 725284721, 3423760262, 16251813715, 77523741208, 371428985796, 1786623827240, 8624669381161, 41769772877288, 202893913979291, 988224403828490, 4825331506973445
Offset: 0

Views

Author

Joerg Arndt, May 26 2018

Keywords

Links

Crossrefs

Cf. A007317, A000699, A088716.

Programs

  • Mathematica
    CoefficientList[Series[(1 - Sqrt[1 - 4*x*(1 + x/(1 - x)^2)]) / (2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 30 2020 *)
  • PARI
    seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1,  1 + a[k]*a[n-k])); a;
};
seq(32)

Formula

G.f. A(x) satisfies: A(x) = 1 + x * (1/(1 - x)^2 + A(x)^2). - Ilya Gutkovskiy, Jun 30 2020
From Vaclav Kotesovec, Jun 30 2020: (Start)
G.f.: (1 - sqrt(1 - 4*x*(1 + x/(1 - x)^2))) / (2*x).
a(n) ~ sqrt(1/r + 2/(1 - r)^3) / (2*sqrt(Pi) * n^(3/2) * r^n), where r = 0.19288682865259090392018... is the real root of the equation -1 + 6*r - 5*r^2 + 4*r^3 = 0. (End)

A316703 O.g.f. A(x) satisfies: [x^n] exp(-n*x*A(x)) * (n+1 - n/A(x)) = 0 for n>=1.

Original entry on oeis.org

1, 1, 3, 11, 48, 231, 1198, 6571, 37708, 224612, 1381047, 8728357, 56520580, 374049962, 2524760084, 17352434291, 121271358844, 860832917836, 6200469605740, 45281350853036, 335040385844140, 2510109678504943, 19031562100423046, 145961670086604701, 1131893950714288692, 8871861944975204172
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.
Compare to: [x^n] exp(-n^2 * x*G(x)) * (n + 1 - n/G(x)) = 0, which holds for n>=1 when G(x) = 1 + x*G(x)^2 + x^2*G(x)*G'(x), the o.g.f. of A088716.

Examples

			O.g.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 48*x^4 + 231*x^5 + 1198*x^6 + 6571*x^7 + 37708*x^8 + 224612*x^9 + 1381047*x^10 + ...
such that A(x) = 1 + x*A(x)^2 + x^2*A(x)^3 + x^3*A(x)^2*A'(x).
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*x*A(x)) * (n+1 - n/A(x)) begins:
n=1: [1, 0, 1, 8, 189, 5024, 173725, 7248744, 358001497, ...];
n=2: [1, 0, 0, 4, 192, 5568, 210880, 9271680, 476620032, ...];
n=3: [1, 0, -3, 0, 117, 4032, 180225, 8532864, 462998025, ...];
n=4: [1, 0, -8, 8, 0, 1856, 122560, 6538752, 384283648, ...];
n=5: [1, 0, -15, 40, -195, 0, 59725, 4165800, 280894425, ...];
n=6: [1, 0, -24, 108, -576, -576, 0, 1897344, 175779072, ...];
n=7: [1, 0, -35, 224, -1323, 1568, -60095, 0, 80665417, ...];
n=8: [1, 0, -48, 400, -2688, 8832, -137600, -1330176, 0, ...];
n=9: [1, 0, -63, 648, -4995, 25056, -268515, -1821528, -66997287, 0, ...]; ...
in which the zero coefficient of x^n is shown as a diagonal of zeros.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 361*x^4/4! + 7701*x^5/5! + 218851*x^6/6! + 7835773*x^7/7! + 339872625*x^8/8! + ...
1/A(x) = 1 - x - 2*x^2 - 6*x^3 - 25*x^4 - 118*x^5 - 612*x^6 - 3382*x^7 - 19639*x^8 - 118618*x^9 + ...
A'(x)/A(x) = 1 + 5*x + 25*x^2 + 141*x^3 + 836*x^4 + 5183*x^5 + 33202*x^6 + 218613*x^7 + 1473064*x^8 + ...

Links

Crossrefs

Cf. A088716.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); m=#A-1; A[#A] = -Vec( exp(-m * x*Ser(A) ) * (m+1 - m/Ser(A)))[#A]/m );A[n+1]}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* Differential equation: */
{a(n) = my(A=1); for(i=1,n, A = 1 + x*A^2 + x^2*A^3 + x^3*A^2*A' + x*O(x^n) ); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

O.g.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 + x^2*A(x)^3 + x^3*A(x)^2*A'(x).
Previous Showing 51-55 of 55 results.

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