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.

Showing 1-10 of 13 results. Next

A301363 G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x)/(1 - 2*x*A(x)/(1 - 3*x*A(x)/(1 - 4*x*A(x)/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 4, 25, 202, 1966, 22306, 289969, 4272934, 70792318, 1308702592, 26791202362, 602762346088, 14795609964448, 393567982759966, 11276489767853569, 346158428070229414, 11331678979354212886, 393967314482937530248, 14495027742943618066030, 562600190990455844759356
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 19 2018

Keywords

Examples

			G.f. A(x) = 1 + x + 4*x^2 + 25*x^3 + 202*x^4 + 1966*x^5 + 22306*x^6 + 289969*x^7 + 4272934*x^8 + ...
The g.f. also satisfies:
A(x) = 1 + x*A(x) + 3*x^2*A(x) + 15*x^3*A(x)^3 + 105*x^4*A(x)^4 + 945*x^5*A(x)^5 + 10395*x^6*A(x)^6 + ... + (2*n)!/(n!*2^n) * x^n * A(x)^n + ...

Links

Crossrefs

Cf. A001147, A088368, A302100, A302535, A302565.

Programs

  • PARI
    /* Continued Fraction */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1,n, A=CF; for(k=0, n, CF = 1/(1 - (n-k+1)*x*A*CF ) )); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Differential Equation */
{a(n) = my(A=1); for(i=0,n, A = 1 + x*(x*A^3)'/(x*A +x^2*O(x^n))'); polcoeff(A,n)}
for(n=0, 30, print1(a(n),", "))

Formula

G.f. A(x) satisfies: A(x) = 1 + x * (x*A(x)^3)' / (x*A(x))'. - Paul D. Hanna, Apr 01 2018
G.f. A(x) satisfies: A(x) = (1/x)*Series_Reversion(x/F(x)) where F(x) = A(x/F(x)) = Sum_{n>=0} (2*n)!/(n!*2^n)*x^n is an o.g.f. of A001147. - Paul D. Hanna, Apr 05 2018
G.f. A(x) satisfies: A(x) = Sum_{n>=0} (2*n)!/(n!*2^n) * x^n * A(x)^n. - Paul D. Hanna, Apr 09 2018
a(n) ~ 2^(n + 1/2) * n^n / exp(n - 1/2). - Vaclav Kotesovec, Jun 18 2019

A302100 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (3*k + 1).

Original entry on oeis.org

1, 1, 5, 41, 449, 6081, 98133, 1846377, 39888353, 977117825, 26839621829, 818332799593, 27443807417569, 1004188344449473, 39809506543659477, 1699473112658002089, 77716022374143303489, 3789578550994707778305, 196255782523222432943109, 10756748528551996006448553, 622036345094017435642828161, 37846075344692579622469742529
Offset: 0

Views

Author

Paul D. Hanna, Apr 09 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 41*x^3 + 449*x^4 + 6081*x^5 + 98133*x^6 + 1846377*x^7 + 39888353*x^8 + 977117825*x^9 + 26839621829*x^10 + ...
such that
A(x) = 1 + x*A(x) + 4*x^2*A(x)^2 + 28*x^3*A(x)^3 + 280*x^4*A(x)^4 + 3640*x^5*A(x)^5 + 58240*x^6*A(x)^6 + ... + x^n * A(x)^n * Product_{k=0..n-1} (3*k + 1) + ...

Links

Crossrefs

Cf. A007559, A088368, A301363, A302535, A302565.

Programs

  • PARI
    /* Series Reversion of Triple Factorials g.f.: */
{a(n) = polcoeff((1/x) * serreverse(x/sum(m=0, n, x^m * prod(k=0,m-1,3*k + 1)) +x^2*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Differential Equation: */
{a(n) = my(A=1); for(i=0, n, A = 1 + x*A^2*(A + 4*x*A')/(x*A +x^2*O(x^n))'); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Continued fraction: */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - floor(3*(n-k+1)/2)*x*A*CF ) )); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (3*k + 1).
(2) A(x) = (1/x)*Series_Reversion( x/F(x) ), where F(x) = Sum_{n>=0} A007559(n)*x^n, the o.g.f. of the triple factorials.
(3) A(x) = 1 + x*A(x)^2 * (A(x) + 4*x*A'(x)) / (A(x) + x*A'(x)).
(4) A(x) = 1/(1 - x*A(x)/(1 - 3*x*A(x)/(1 - 4*x*A(x)/(1 - 6*x*A(x)/(1 - 7*x*A(x)/(1 - 9*x*A(x)/(1 - 10*x*A(x)/(1 - ...)))))))), a continued fraction.
a(n) ~ sqrt(2*Pi) * 3^n * n^(n - 1/6) / (Gamma(1/3) * exp(n - 1/3)). - Vaclav Kotesovec, Jun 18 2019

A302535 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1).

Original entry on oeis.org

1, 1, 6, 61, 846, 14746, 310016, 7665141, 218827766, 7106293246, 259169817316, 10497928495506, 467768758203676, 22739720141372196, 1197560448125948596, 67910602688355999461, 4125144974025630599846, 267199960610924528490486, 18382741943990196237909476, 1338585578875261292134492646, 102848696213697953204782043556
Offset: 0

Views

Author

Paul D. Hanna, Apr 09 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 61*x^3 + 846*x^4 + 14746*x^5 + 310016*x^6 + 7665141*x^7 + 218827766*x^8 + 7106293246*x^9 + 259169817316*x^10 + ...
such that
A(x) = 1 + x*A(x) + 5*x^2*A(x)^2 + 45*x^3*A(x)^3 + 585*x^4*A(x)^4 + 9945*x^5*A(x)^5 + 208845*x^6*A(x)^6 + ... + x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1) + ...

Links

Crossrefs

Cf. A007696, A088368, A301363, A302100, A302565.

Programs

  • PARI
    /* Series Reversion of Quartic Factorials g.f.: */
{a(n) = polcoeff((1/x) * serreverse(x/sum(m=0, n, x^m*prod(k=1,m-1,4*k + 1))+x^2*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Differential Equation: */
{a(n) = my(A=1); for(i=0, n, A = 1 + x*A^2*(A + 5*x*A')/(x*A +x^2*O(x^n))'); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Continued fraction: */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - floor(4*floor(3*(n-k+1)/2)/3)*x*A*CF ) )); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1).
(2) A(x) = (1/x)*Series_Reversion( x/F(x) ), where F(x) = Sum_{n>=0} A007696(n)*x^n, the o.g.f. of the quartic factorials.
(3) A(x) = 1 + x*A(x)^2 * (A(x) + 5*x*A'(x)) / (A(x) + x*A'(x)).
(4) A(x) = 1/(1 - x*A(x)/(1 - 4*x*A(x)/(1 - 5*x*A(x)/(1 - 8*x*A(x)/(1 - 9*x*A(x)/(1 - 12*x*A(x)/(1 - 13*x*A(x)/(1 - ...)))))))), a continued fraction.
a(n) ~ sqrt(Pi) * 2^(2*n + 1/2) * n^(n - 1/4) / (Gamma(1/4) * exp(n - 1/4)). - Vaclav Kotesovec, Jun 18 2019

A302565 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (5*k + 1).

Original entry on oeis.org

1, 1, 7, 85, 1429, 30517, 792007, 24293389, 862902745, 34918162057, 1587910815271, 80217252865861, 4457823231346717, 270261899977497325, 17749585402744292215, 1255201826997862952845, 95083758340337074058545, 7680863233559647281837265, 659040900304099125516970375, 59855299015030039092312638965
Offset: 0

Views

Author

Paul D. Hanna, Apr 09 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 85*x^3 + 1429*x^4 + 30517*x^5 + 792007*x^6 + 24293389*x^7 + 862902745*x^8 + 34918162057*x^9 + ...
such that
A(x) = 1 + x*A(x) + 6*x^2*A(x)^2 + 66*x^3*A(x)^3 + 1056*x^4*A(x)^4 + 22176*x^5*A(x)^5 + ... + x^n*A(x)^n * Product_{k=0..n-1} (5*k + 1) + ...

Links

Crossrefs

Cf. A008548, A088368, A301363, A302100, A302535.

Programs

  • PARI
    /* Series Reversion of Quintuple Factorials g.f.: */
{a(n) = polcoeff((1/x) * serreverse(x/sum(m=0, n, x^m * prod(k=0, m-1, 5*k + 1)) +x^2*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Differential Equation: */
{a(n) = my(A=1); for(i=0, n, A = 1 + x*A^2*(A + 6*x*A')/(x*A +x^2*O(x^n))'); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Continued fraction: */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - floor(5*floor(3*(n-k+1)/2)/3)*x*A*CF ) )); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (5*k + 1).
(2) A(x) = (1/x)*Series_Reversion( x/F(x) ), where F(x) = Sum_{n>=0} A008548(n)*x^n, the o.g.f. of the quintuple factorials.
(3) A(x) = 1 + x*A(x)^2 * (A(x) + 6*x*A'(x)) / (A(x) + x*A'(x)).
(4) A(x) = 1/(1 - x*A(x)/(1 - 5*x*A(x)/(1 - 6*x*A(x)/(1 - 10*x*A(x)/(1 - 11*x*A(x)/(1 - 15*x*A(x)/(1 - 16*x*A(x)/(1 - ...)))))))), a continued fraction.
a(n) ~ sqrt(2*Pi) * 5^n * n^(n - 3/10) / (Gamma(1/5) * exp(n - 1/5)). - Vaclav Kotesovec, Jun 18 2019

A307441 G.f. A(x) satisfies: A(x) = Sum_{k>=0} k!*x^k*A(x)^k/(1 - x)^(k+1).

Original entry on oeis.org

1, 2, 7, 35, 216, 1527, 11927, 101056, 920055, 8960343, 93202418, 1035640333, 12305625141, 156513872514, 2131781868823, 31077520424879, 484157377851360, 8040920113043655, 141937291242762263, 2654252437895865112, 52412046969340405371, 1089506079309378596823
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 08 2019

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 35*x^3 + 216*x^4 + 1527*x^5 + 11927*x^6 + 101056*x^7 + 920055*x^8 + 8960343*x^9 + 93202418*x^10 + ...

Links

Crossrefs

Cf. A000522, A088368, A307442, A307443, A307444.

Programs

  • Mathematica
    terms = 22; A[] = 1; Do[A[x] = Sum[k! x^k A[x]^k/(1 - x)^(k + 1), {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 22; A[] = 1; Do[A[x] = Sum[x^j Sum[k! Binomial[j, k] A[x]^k, {k, 0, j}], {j, 0, i}] + O[x]^i, {i, 1, terms}]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = Sum_{j>=0} x^j * Sum_{k=0..j} k!*binomial(j,k)*A(x)^k.
a(n) ~ exp(3) * n!. - Vaclav Kotesovec, Apr 10 2019

A307442 G.f. A(x) satisfies: A(x) = Sum_{k>=0} k!*x^k*A(x)^k/(1 - x*A(x))^(k+1).

Original entry on oeis.org

1, 2, 9, 54, 379, 2948, 24736, 220622, 2074775, 20491386, 212312349, 2310232488, 26473612772, 320735694048, 4126350096188, 56601987176510, 830233489763775, 13036492313617494, 218958840306428947, 3924128327446669670, 74779561501535316579, 1509296316416028136188
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 08 2019

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 54*x^3 + 379*x^4 + 2948*x^5 + 24736*x^6 + 220622*x^7 + 2074775*x^8 + 20491386*x^9 + 212312349*x^10 + ...

Crossrefs

Cf. A000522, A088368, A307441, A307443, A307444.

Programs

  • Mathematica
    terms = 22; CoefficientList[1/x InverseSeries[Series[x/(1 + Sum[Floor[Exp[1] k!] x^k, {k, 1, terms}]), {x, 0, terms}], x], x]
terms = 22; A[] = 1; Do[A[x] = Sum[k! x^k A[x]^k/(1 - x A[x])^(k + 1), {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 22; A[] = 1; Do[A[x] = 1 + Sum[Floor[Exp[1] k!] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = Sum_{k>=0} A000522(k)*x^k*A(x)^k.
G.f.: A(x) = (1/x)*Series_Reversion(x/Sum_{k>=0} A000522(k)*x^k).
a(n) ~ exp(3) * n!. - Vaclav Kotesovec, Apr 10 2019

A307443 G.f. A(x) satisfies: A(x) = Sum_{k>=0} k!*x^k*A(x)^k/(1 + x)^(k+1).

Original entry on oeis.org

1, 0, 1, 3, 14, 73, 439, 2986, 22849, 195639, 1864072, 19639587, 227216485, 2866190328, 39155468153, 575750407431, 9063067630294, 152007287492665, 2705337486885751, 50909087031293746, 1009776468826520181, 21052688394533433215, 460223336063328374304, 10525518902412521320567
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 08 2019

Keywords

Examples

			G.f.: A(x) =  1 + x^2 + 3*x^3 + 14*x^4 + 73*x^5 + 439*x^6 + 2986*x^7 + 22849*x^8 + 195639*x^9 + 1864072*x^10 + ...

Crossrefs

Cf. A000166, A088368, A307441, A307442, A307444.

Programs

  • Mathematica
    terms = 24; A[] = 1; Do[A[x] = Sum[k! x^k A[x]^k/(1 + x)^(k + 1), {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 24; A[] = 1; Do[A[x] = Sum[x^j Sum[(-1)^(j - k) k! Binomial[j, k] A[x]^k, {k, 0, j}], {j, 0, i}] + O[x]^i, {i, 1, terms}]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = Sum_{j>=0} x^j * Sum_{k=0..j} (-1)^(j-k)*k!*binomial(j,k)*A(x)^k.
a(n) ~ exp(-1) * n!. - Vaclav Kotesovec, Apr 10 2019

A307444 G.f. A(x) satisfies: A(x) = Sum_{k>=0} k!*x^k*A(x)^k/(1 + x*A(x))^(k+1).

Original entry on oeis.org

1, 0, 1, 2, 11, 54, 336, 2330, 18359, 161660, 1580853, 17031728, 200718372, 2569989304, 35531288796, 527506796282, 8368806193151, 141271243571640, 2527897717923387, 47789579768358498, 951677263953890739, 19910429474370487166, 436589745454529328720, 10012315468481417357976
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 08 2019

Keywords

Examples

			G.f.: A(x) =  1 + x^2 + 2*x^3 + 11*x^4 + 54*x^5 + 336*x^6 + 2330*x^7 + 18359*x^8 + 161660*x^9 + 1580853*x^10 + ...

Crossrefs

Cf. A000166, A088368, A307441, A307442, A307443.

Programs

  • Mathematica
    terms = 24; CoefficientList[1/x InverseSeries[Series[x/Sum[Subfactorial[k] x^k, {k, 0, terms}], {x, 0, terms}], x], x]
terms = 24; A[] = 1; Do[A[x] = Sum[k! x^k A[x]^k/(1 + x A[x])^(k + 1), {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 24; A[] = 1; Do[A[x] = Sum[Subfactorial[k] x^k A[x]^k, {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = Sum_{k>=0} A000166(k)*x^k*A(x)^k.
G.f.: A(x) = (1/x)*Series_Reversion(x/Sum_{k>=0} A000166(k)*x^k).
a(n) ~ exp(-1) * n!. - Vaclav Kotesovec, Apr 10 2019

A193092 Augmentation of the triangular array P given by p(n,k)=k! for 0<=k<=n. See Comments.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 13, 1, 4, 12, 32, 69, 1, 5, 18, 58, 173, 421, 1, 6, 25, 92, 321, 1058, 2867, 1, 7, 33, 135, 523, 1977, 7159, 21477, 1, 8, 42, 188, 790, 3256, 13344, 53008, 175769, 1, 9, 52, 252, 1134, 4986, 21996, 97956, 427401, 1567273
Offset: 0

Views

Author

Clark Kimberling, Jul 30 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding W=A193092, we have w(n,n)=A088368.

Examples

			First 7 rows:
1
1...1
1...2....3
1...3....7.....13
1...4....12....32....69
1...5....18....58....173...421
1...6....25....92....321...1058...2867
The matrix method described at A193091 shows that row 3 arises from row 2 as the matrix product
............. (1...1...2...4)
(1...2...3) * (0...1...1...2) = (1...3...7...13)
............. (0...0...1...1).
The equivalent polynomial substitution method:
x^2+2x+3 -> (x^3+x^2+2x+4)+2(x^2+x+2)+3(x+1)= x^3+3x^2+7x+13.

Crossrefs

Cf. A088368, A193091.

Programs

  • Mathematica
    p[n_, k_] := k!
Table[p[n, k], {n, 0, 5}, {k, 0, n}]
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193092 *)
Flatten[Table[v[n], {n, 0, 8}]]

A198916 G.f. satisfies: A(x) = Sum_{n>=0} n!*x^n*A(x)^(2*n).

Original entry on oeis.org

1, 1, 4, 23, 158, 1212, 10058, 88811, 826982, 8085950, 82922624, 893003234, 10129641140, 121552747370, 1550460365622, 21115793548491, 308004022741254, 4817224946243142, 80703099826887368, 1444218797390453282, 27501426760092853796, 554910390616969332656
Offset: 0

Views

Author

Paul D. Hanna, Oct 31 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 158*x^4 + 1212*x^5 + 10058*x^6 +...
where
A(x) = 1 + x*A(x)^2 + 2!*x^2*A(x)^4 + 3!*x^3*A(x)^6 + 4!*x^4*A(x)^8 +...

Crossrefs

Cf. A088368.

Programs

  • PARI
    {a(n)=polcoeff((1/x*serreverse(x/sum(m=0, n, m!*x^m+x^2*O(x^n))^2))^(1/2), n)}
  • PARI
    /* Recursive continued fraction: */
{a(n)=local(A=1+x,CF=1+x*O(x^(n+2))); for(i=1,n, for(k=1, n+1, CF=1/(1-((n-k+1)\2+1)*x*A^2*CF));A=CF); polcoeff(A,n)}
  • PARI
    {a(n)=polcoeff(sum(k=0,n,k!*x^k +x*O(x^n))^(2*n+1)/(2*n+1),n)}

Formula

G.f.: A(x) = sqrt((1/x)*Series_Reversion( x/[Sum_{n>=0} n!*x^n]^2 )).
G.f. satisfies: A(x) = 1/(1 - x*A(x)^2/(1 - x*A(x)^2/(1 - 2*x*A(x)^2/(1 - 2*x*A(x)^2/(1 - 3*x*A(x)^2/(1 - 3*x*A(x)^2/(1 - 4*x*A(x)^2/(1 - ...)))))))), a recursive continued fraction.
G.f. satisfies: A(x/F(x)^2) = F(x) where F(x) = Sum_{n>=0} n!*x^n.
a(n) = [x^n] (Sum_{k>=0} k!*x^k)^(2*n+1) / (2*n+1).
Showing 1-10 of 13 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.