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 1-9 of 9 results.

A161633 E.g.f. satisfies A(x) = 1/(1 - x*exp(x*A(x))).

Original entry on oeis.org

1, 1, 4, 27, 268, 3525, 57966, 1146061, 26500552, 702069129, 20974309210, 697754762001, 25584428686620, 1025230366195789, 44579963354153878, 2090676600895922565, 105191995364927688976, 5652501986238910061073, 323083811850594613809714, 19573120681427758058921881
Offset: 0

Views

Author

Paul D. Hanna, Jun 18 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 268*x^4/4! + 3525*x^5/5! +...
exp(x*A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...

Links

Crossrefs

Cf. A006153, A161630 (e.g.f. = exp(x*A(x))), A213644, A364980, A364981.
Cf. A366232, A366233, A366234, A366235.

Programs

  • Mathematica
    Flatten[{1,Table[n!*Sum[Binomial[n+1,k]/(n+1) * k^(n-k)/(n-k)!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jan 10 2014 *)
  • PARI
    a(n,m=1)=n!*sum(k=0,n,binomial(n+m,k)*m/(n+m)*k^(n-k)/(n-k)!)

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*A(x)*exp(x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(x)) ).
(3) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-1)*x*A(x)) for all fixed nonnegative m.
a(n) = n! * Sum_{k=0..n} binomial(n+1,k)/(n+1) * k^(n-k)/(n-k)!.
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then a(n,m) = n! * Sum_{k=0..n} binomial(n+m,k)*m/(n+m) * k^(n-k)/(n-k)!.
a(n) ~ n^(n-1) * c * ((c-1)*c)^(n+1/2) / (sqrt(2*c-1) * exp(n)), where c = 1 + 1/(2*LambertW(1/2)) = 2.4215299358831166... - Vaclav Kotesovec, Jan 10 2014

A161635 E.g.f. satisfies A(x) = exp( x/(1 - x*A(x))^2 ).

Original entry on oeis.org

1, 1, 5, 43, 553, 9501, 204961, 5330599, 162432593, 5677941817, 224018814241, 9848702243931, 477481361216377, 25309471236379669, 1456206709854725921, 90387017392004356591, 6020486941130334199201, 428348710658269120403313
Offset: 0

Views

Author

Paul D. Hanna, Jun 19 2009

Keywords

Examples

			E.g.f: A(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 553*x^4/4! + 9501*x^5/5! +...
log(A(x))/x = 1 + 2*x*A(x) + 3*x^2*A(x)^2 + 4*x^3*A(x)^3 + 5*x^4*A(x)^4 +...

Links

Crossrefs

Cf. A161630, A364980.

Programs

  • Mathematica
    Flatten[{1,Table[n!*Sum[(n-k+1)^(k-1)/k! * Binomial[n+k-1,n-k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jan 10 2014 *)
FindRoot[{2*s*r^2 == (1-r*s)^3, r == Log[s]*(1-r*s)^2},{r,1/2},{s,1}, WorkingPrecision->50] (* program for numerical values of constants r and s, Vaclav Kotesovec, Jan 10 2014 *)
  • PARI
    {a(n,m=1)=if(n==0,1,sum(k=0,n,n!/k!*m*(n-k+m)^(k-1)*binomial(n+k-1,n-k)))}
  • PARI
    {a(n,m=1)=my(A=1+x+x*O(x^n));for(i=1,n,A=exp(x/(1-x*A)^2));n!*polcoeff(A^m,n)}

Formula

a(n) = n!*Sum_{k=0..n} (n-k+1)^(k-1)/k! * C(n+k-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = n!*Sum_{k=0..n} m*(n-k+m)^(k-1)/k! * C(n+k-1,n-k).
...
E.g.f.: A(x) = (1/x)*Series_Reversion[ (1-x)^2*LambertW(x/(1-x)^2) ].
a(n) ~ sqrt(s*(1+r*s)/(3+2*r-6*r*s+3*r^2*s^2)) * n^(n-1) / (exp(n) * r^n), where r = 0.219092370374602436272454670371..., s = 1.952248277910295452167538973654... are the roots of the equations 2*s*r^2 = (1-r*s)^3 and r = log(s) * (1-r*s)^2. - Vaclav Kotesovec, Jan 10 2014

A364981 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x*A(x)^3).

Original entry on oeis.org

1, 1, 4, 39, 580, 11685, 298566, 9248701, 336886936, 14112113049, 668422303210, 35325208755441, 2060811941835780, 131547166492534117, 9120279070776381886, 682489450793082237285, 54828316394224735284016, 4706545644403274325580593
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A006153, A161633, A364938, A364980.

Programs

  • Mathematica
    Join[{1}, Table[n! * Sum[k^(n-k) * Binomial[3*n-2*k+1,k] / ((3*n-2*k+1)*(n-k)!), {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Nov 18 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3*n-2*k+1, k)/((3*n-2*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*n-2*k+1,k)/( (3*n-2*k+1)*(n-k)! ).
a(n) ~ sqrt((1 + r*s^3)/(12*s + 9*r*s^4)) * n^(n-1) / (exp(n) * r^(n + 1/2)), where r = 0.1811100305436879929789759231994897963241226689807... and s = 1.522012903517407628213363540403002787906223513104... are real roots of the system of equations 1 + exp(r*s^3)*r*s = s, 3*r*s^3*(s-1) = 1. - Vaclav Kotesovec, Nov 18 2023

A381378 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cos(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 2, 3, -48, -1135, -18240, -231637, -1356544, 53849889, 3026119680, 100808786419, 2429052865536, 26284690243825, -1539261873164288, -140633348417624805, -7196339681250508800, -258335768147494234303, -4225401456668904259584, 307227604973975435785571
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A364980, A381376, A381382, A381384.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n-k+1, k)/(2*n-k+1)*I^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(2*n-k+1,k)/(2*n-k+1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381382 E.g.f. A(x) satisfies A(x) = 1/( 1 - sinh(x * A(x)^2) / A(x)^2 ).

Original entry on oeis.org

1, 1, 2, 7, 48, 541, 7600, 120891, 2178176, 45053401, 1065957888, 28344376303, 831973593088, 26647344263541, 925300511922176, 34668496386129763, 1394928344160731136, 59986286728056665905, 2744940504174063714304, 133158543838350039763671
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2025

Keywords

Crossrefs

Cf. A364980, A381376, A381378, A381384.
Cf. A136630.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n-k+1, k)/(2*n-k+1)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(2*n-k+1,k)/(2*n-k+1) * A136630(n,k).

A381384 E.g.f. A(x) satisfies A(x) = 1/( 1 - sin(x * A(x)^2) / A(x)^2 ).

Original entry on oeis.org

1, 1, 2, 5, 0, -299, -5840, -90791, -1210496, -11174519, 71397888, 8367496301, 327020705792, 9709296136541, 226223975684096, 2946493117173761, -87437164233621504, -9675847870039338095, -535455805780063748096, -22518479178045130002731, -706013052362778282033152
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2025

Keywords

Crossrefs

Cf. A364980, A381376, A381378, A381382.
Cf. A136630.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n-k+1, k)/(2*n-k+1)*I^(n-k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(2*n-k+1,k)/(2*n-k+1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A377541 E.g.f. satisfies A(x) = 1/(1 - x * exp(x*A(x)))^2.

Original entry on oeis.org

1, 2, 10, 90, 1184, 20650, 450252, 11803526, 361892848, 12712357170, 503564718260, 22212233618542, 1079909444635848, 57379354040049002, 3308238701451609772, 205715613407117613270, 13724187813695296374752, 977841609869801208944482, 74108335568947966714172004
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2024

Keywords

Crossrefs

Cf. A161633, A377545, A377551.
Cf. A364980.

Programs

  • PARI
    a(n) = 2*n!*sum(k=0, n, k^(n-k)*binomial(2*n-k+2, k)/((2*n-k+2)*(n-k)!));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364980.
a(n) = 2 * n! * Sum_{k=0..n} k^(n-k) * binomial(2*n-k+2,k)/( (2*n-k+2)*(n-k)! ).

A377550 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x*A(x)^4).

Original entry on oeis.org

1, 1, 4, 45, 772, 17865, 525966, 18794881, 790175128, 38221092657, 2091074167450, 127675964340441, 8606833626646740, 634928943628432921, 50878715440232312374, 4400937219238706030865, 408700742920092110904496, 40558224679468186878237153, 4283310197644529184427059378
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2024

Keywords

Crossrefs

Cf. A161633, A364980, A364981.
Cf. A377551, A377552.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*n-3*k+1, k)/((4*n-3*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*n-3*k+1,k)/( (4*n-3*k+1)*(n-k)! ).

A377549 E.g.f. satisfies A(x) = 1 + x*A(x)^5*exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 12, 285, 10444, 520465, 32882406, 2519264797, 227003238792, 23526134771553, 2757165645132010, 360564513170510341, 52053350012338720332, 8222888925567102799441, 1410913077291231960911934, 261306906300110395598900685, 51955790654759866661097707536
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2024

Keywords

Crossrefs

Cf. A364980, A364982, A364985, A365176.
Cf. A377547.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*n+3*k+1,k)/( (2*n+3*k+1)*(n-k)! ).
Showing 1-9 of 9 results.

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