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.

Previous Showing 11-20 of 23 results. Next

A360601 E.g.f. satisfies A(x) = exp(x*A(x)^2) / (1-x).

Original entry on oeis.org

1, 2, 13, 166, 3265, 87306, 2957509, 121400350, 5857287937, 324884241874, 20370279663901, 1424790170536470, 109990236302275201, 9289460282062082266, 852049115732672006101, 84345608594930495005966, 8962937531710834906989313, 1017655033307013508626619554
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2023

Keywords

Links

Crossrefs

Cf. A352410, A360609.
Cf. A052750, A352448.
Cf. A370875.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(lambertw(-2*x/(1-x)^2)/(-2*x))))

Formula

E.g.f.: sqrt(LambertW( -2*x/(1-x)^2 ) / (-2*x)).
a(n) ~ sqrt(1 + 2*exp(-1) - sqrt(1 + 2*exp(-1))) * n^(n-1) / (2 * (sqrt(1 + 2*exp(-1)) - 1)^(3/2) * exp(2*n + 1/2) * (1 + exp(-1) - sqrt(1 + 2*exp(-1)))^n). - Vaclav Kotesovec, Mar 06 2023
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n+k,n-k)/k!. - Seiichi Manyama, Mar 09 2024

A384617 E.g.f. A(x) satisfies A(x) = exp( x/A(-x*A(x))^2 ).

Original entry on oeis.org

1, 1, 5, 13, -63, -2279, -51167, -423387, 13717889, 885044593, 37051519041, 779965433149, -14179999608959, -2798466635425239, -224720509492366495, -11148988922254048619, -300176114650473574143, 18804123010954180467937, 4351564646569010083711105
Offset: 0

Views

Author

Seiichi Manyama, Jun 10 2025

Keywords

Crossrefs

Column k=1 of A384808.
Cf. A052750, A213110, A213111, A384809, A384810.
Cf. A213108.

Programs

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

Formula

See A384808.

A384809 E.g.f. A(x) satisfies A(x) = exp( x/A(-x*A(x)^2)^2 ).

Original entry on oeis.org

1, 1, 5, 25, 153, -799, -82787, -2990343, -98020367, -2473062911, -22379003019, 3535310560409, 426542722323721, 33942691393940577, 2320589389274335117, 131491185267395291641, 4583444982950062321377, -254657491559719266483967, -86887910247671284788294683
Offset: 0

Views

Author

Seiichi Manyama, Jun 10 2025

Keywords

Crossrefs

Column k=1 of A384811.
Cf. A052750, A213110, A213111, A384617, A384810.

Programs

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

Formula

See A384811.

A384810 E.g.f. A(x) satisfies A(x) = exp( x/A(-x*A(x)^3)^2 ).

Original entry on oeis.org

1, 1, 5, 37, 417, 4761, 33313, -1509339, -135791359, -8149132943, -455269648959, -24532196772291, -1260399381304511, -56411711489070807, -1357347436103060191, 146282852689561868821, 35003916010171558562817, 5112183093788001812407521, 647998390863196992450043777
Offset: 0

Views

Author

Seiichi Manyama, Jun 10 2025

Keywords

Crossrefs

Column k=1 of A384813.
Cf. A052750, A213110, A213111, A384617, A384809.

Programs

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

Formula

See A384813.

A058927 Numerators of series related to triangular cacti.

Original entry on oeis.org

1, 1, 5, 49, 243, 14641, 371293, 253125, 410338673, 16983563041, 1400846643, 41426511213649, 95367431640625, 617673396283947, 10260628712958602189, 756943935220796320321, 7474615974418932603, 827909024473876953125, 456487940826035155404146917, 510798409623548623605717
Offset: 0

Views

Author

N. J. A. Sloane, Jan 12 2001

Keywords

Comments

From L. Edson Jeffery, Jan 09 2012: (Start)
The reference [Bergeron, et al.] lists the first few terms of the relevant series as S(x) = x + (1/2)*x^3 + (5/8)*x^5 + (49/48)*x^7 + (243/128)*x^9 + ..., from which the numerators were taken for this sequence and the denominators for A058928. This leads to the following
Conjecture: S(x) = Sum_{n>=0} ((2*n+1)^(n-1)/(n!*2^n))*x^(2*n+1) = (A052750(n)/A000165(n))*x^(2*n+1). Letting D_n be the set of divisors of n! and d_n = max(k in D_n : k | (2*n+1)^(n-1)), then a(n)=A052750(n)/d_n. (End)
The above conjecture is correct and follows from formula given in A034940 for the number of rooted labeled triangular cacti with 2n+1 nodes. - Andrew Howroyd, Aug 30 2018

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 307.

Links

Crossrefs

Cf. A000165, A034940, A052750, A058928.

Programs

  • PARI
    a(n)={numerator((2*n+1)^(n-1)/(2^n*n!))} \\ Andrew Howroyd, Aug 30 2018

Formula

G.f.: A(x) satisfies A(x)=exp(x*A(x)^2). - Vladimir Kruchinin, Feb 09 2013
a(n) = numerator(A034940(n)/(2*n+1)!) = numerator((2*n+1)^(n-1)/(2^n*n!)). - Andrew Howroyd, Aug 30 2018

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010
Terms a(12) and beyond from Andrew Howroyd, Aug 30 2018

A058928 Denominators of series related to triangular cacti.

Original entry on oeis.org

1, 2, 8, 48, 128, 3840, 46080, 14336, 10321920, 185794560, 6553600, 81749606400, 78479622144, 209924915200, 1428329123020800, 42849873690624000, 170993385472000, 7611536747003904, 1678343852714360832000, 747740921331712000, 2551082656125828464640000
Offset: 0

Views

Author

N. J. A. Sloane, Jan 12 2001

Keywords

Comments

From L. Edson Jeffery, Jan 09 2012: (Start)
The reference [Bergeron, et al.] lists the first few terms of the relevant series as S(x) = x + (1/2)*x^3 + (5/8)*x^5 + (49/48)*x^7 + (243/128)*x^9 + ..., from which the denominators were taken for this sequence and the numerators for A058927. This leads to the following
Conjecture: S(x) = Sum_{n>=0} ((2*n+1)^(n-1)/(n!*2^n))*x^(2*n+1) = (A052750(n)/A000165(n))*x^(2*n+1). Letting D_n be the set of divisors of n! and d_n = max(k in D_n : k | (2*n+1)^(n-1)), then a(n)=A000165(n)/d_n. (End)
The above conjecture is correct and follows from formula given in A034940 for the number of rooted labeled triangular cacti with 2n+1 nodes. - Andrew Howroyd, Aug 30 2018

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 307.

Links

Crossrefs

Cf. A000165, A034940, A052750, A058927.

Programs

  • PARI
    a(n)={denominator((2*n+1)^(n-1)/(2^n*n!))} \\ Andrew Howroyd, Aug 30 2018

Formula

a(n) = denominator(A034940(n)/(2*n+1)!) = denominator((2*n+1)^(n-1)/(2^n*n!)). - Andrew Howroyd, Aug 30 2018

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010
Terms a(12) and beyond from Andrew Howroyd, Aug 30 2018

A370262 Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k.

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 1, 14, 49, 49, 1, 30, 243, 729, 729, 1, 55, 847, 5324, 14641, 14641, 1, 91, 2366, 26364, 142805, 371293, 371293, 1, 140, 5670, 101250, 928125, 4556250, 11390625, 11390625, 1, 204, 12138, 324258, 4593655, 36916282, 168962983, 410338673, 410338673
Offset: 0

Views

Author

Peter Bala, Mar 12 2024

Keywords

Comments

The table entries are integers since a(n, k) := binomial(n+k, n-k)/(2*k + 1) * (2*n + 1) gives the entries of the transpose of triangle A082985.

Examples

			Triangle begins
 n\k | 0    1      2       3        4        5        6
 - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   0 | 1
   1 | 1    1
   2 | 1    5      5
   3 | 1   14     49      49
   4 | 1   30    243     729      729
   5 | 1   55    847    5324    14641    14641
   6 | 1   91   2366   26364   142805   371293   371293
  ...

Crossrefs

A371697 (row sums), A052750 (main diagonal and subdiagonal), A000330 (column 1).
Cf. A008310, A082985, A258708, A370259, A370260.

Programs

  • Maple
    seq(seq(binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k, k = 0..n), n = 0..10);
  • Mathematica
    Table[Binomial[n + k, n - k] / (2*k + 1) * (2*n + 1)^k, {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 17 2024 *)

Formula

n-th row polynomial R(n, x) = Sum_{k = 0..n} T(n, k)*x^k = sqrt( 2* Sum_{k = 0..2*n} (2*n + 1)^(k-1) *binomial(2*n+k+2, 2*k+2)/(2*n + k + 2) * x^k ).
R(n, x)^2 = 2/(x*(2*n + 1)^3) * ( ChebyshevT(2*n+1, 1 + (2*n+1)*x/2) - 1 ).
R(n, 2) = A370260(n).

A377347 E.g.f. satisfies A(x) = 1 + (exp(x*A(x)^2) - 1)/A(x)^2.

Original entry on oeis.org

1, 1, 1, 7, 41, 391, 4509, 62847, 1038001, 19580071, 418681877, 9973993855, 262293996777, 7545559829991, 235715629493005, 7946944965054271, 287592204406672481, 11120005819664145895, 457514133092462477253, 19957535405566629526335, 920056233384401619083545
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2024

Keywords

Crossrefs

Cf. A052750, A367134, A367180, A377330.
Cf. A377326, A377348.

Programs

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

Formula

a(n) = Sum_{k=0..floor((2*n+1)/3)} (2*n-2*k)!/(2*n-3*k+1)! * Stirling2(n,k).

A362774 E.g.f. satisfies A(x) = exp( x * (1+x)^2 * A(x)^2 ).

Original entry on oeis.org

1, 1, 9, 115, 2265, 59701, 1981513, 79441167, 3736418801, 201790517833, 12309193580841, 837132560820139, 62809405894333321, 5154060532188515325, 459202970647825870313, 44146740571635016905991, 4555272678073789024849377, 502153774773932684443210513
Offset: 0

Views

Author

Seiichi Manyama, May 02 2023

Keywords

Links

Crossrefs

Cf. A361278, A362772.
Cf. A052750, A360547, A362776.

Programs

  • Maple
    A362774 := proc(n)
    n!*add((2*k+1)^(k-1) * binomial(2*k,n-k)/k!,k=0..n) ;
end proc:
seq(A362774(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*(1+x)^2)/2)))

Formula

E.g.f.: exp( -LambertW(-2*x * (1+x)^2)/2 ).
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(2*k,n-k)/k!.

A377330 E.g.f. satisfies A(x) = 1 + A(x)^2 * (exp(x*A(x)^2) - 1).

Original entry on oeis.org

1, 1, 9, 163, 4537, 171451, 8206517, 476071275, 32469361617, 2546397256651, 225784275815485, 22336278201427675, 2439097416667718297, 291422424985108052091, 37817207428965579915333, 5296739332085114983427083, 796419825874139713780172449, 127955324543685857975407200235
Offset: 0

Views

Author

Seiichi Manyama, Oct 25 2024

Keywords

Crossrefs

Cf. A052750, A367134, A367180.

Programs

  • PARI
    a(n) = sum(k=0, n, (2*n+2*k)!/(2*n+k+1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (2*n+2*k)!/(2*n+k+1)! * Stirling2(n,k).
Previous Showing 11-20 of 23 results. Next

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

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