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 11-19 of 19 results.

A052781 G.f. A(x) satisfies: A(x) = exp( Sum_{k>=1} A(x^k)^5 * x^k / k ).

Original entry on oeis.org

1, 1, 6, 46, 421, 4191, 44322, 487662, 5527722, 64091887, 756590138, 9062397539, 109866112785, 1345528776005, 16622049264020, 206882949204038, 2591780764974800, 32656149762325321, 413563728245999232, 5261307475883227222, 67207527369932430625
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Appears to be the Euler transform of A052788. - Falk Hüffner, Dec 03 2015
Old name was: A simple grammar.

Links

Crossrefs

Cf. A005750, A052751, A052773.
Cf. A052788.

Programs

  • Maple
    spec := [S,{B=Prod(Z,S,S,S,S,S),S=Set(B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Formula

G.f. A(x) satisfies: A(x) = exp( Sum_{k>=1} A(x^k)^5 * x^k / k ). - Ilya Gutkovskiy, May 26 2023

Extensions

New name from Ilya Gutkovskiy, May 26 2023

A063688 Number of 2-trees rooted at a triangle.

Original entry on oeis.org

1, 1, 3, 10, 39, 164, 746, 3474, 16658, 81166, 401169, 2004517, 10110757, 51402250, 263133142, 1355126922, 7016115632, 36498130908, 190673015083, 999932115039, 5262094054524, 27779114013628, 147072756065567, 780722981065006
Offset: 1

Views

Author

Vladeta Jovovic, Aug 22 2001

Keywords

References

  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 76, Eq. (3.5.13).

Crossrefs

Cf. A005750, A058870, A058866, A054581.

A363293 G.f. A(x) satisfies: A(x) = x * exp( A(x)^2/x - A(-x^2)^2/(2*x^2) + A(x^3)^2/(3*x^3) - A(-x^4)^2/(4*x^4) + ... ).

Original entry on oeis.org

1, 1, 2, 7, 26, 101, 412, 1756, 7692, 34350, 155980, 718312, 3345890, 15735091, 74613107, 356348561, 1712593184, 8276207120, 40192085383, 196045684833, 960042529894, 4718201036195, 23263440797758, 115042992517035, 570463195069614, 2835840294969867, 14129895469191476
Offset: 1

Views

Author

Ilya Gutkovskiy, May 26 2023

Keywords

Crossrefs

Cf. A005750, A005754, A049075, A363294.

Programs

  • Mathematica
    nmax = 27; A[] = 0; Do[A[x] = x Exp[-Sum[A[-(-x)^k]^2/(k (-x)^k), {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

A363294 G.f. A(x) satisfies: A(x) = x * exp( A(x)^2/x + A(-x^2)^2/(2*x^2) + A(x^3)^2/(3*x^3) + A(-x^4)^2/(4*x^4) + ... ).

Original entry on oeis.org

1, 1, 3, 10, 37, 154, 676, 3053, 14187, 67459, 326241, 1599480, 7933272, 39736160, 200700204, 1021052197, 5227501077, 26912956631, 139244637915, 723631840568, 3775598797694, 19770494002049, 103865161431895, 547291750362216, 2891718659119578, 15317429567883000
Offset: 1

Views

Author

Ilya Gutkovskiy, May 26 2023

Keywords

Crossrefs

Cf. A005750, A005754, A045648, A363293.

Programs

  • Mathematica
    nmax = 26; A[] = 0; Do[A[x] = x Exp[Sum[A[-(-x)^k]^2/(k x^k), {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

A363385 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^2 / k ).

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 4, 11, 14, 29, 66, 115, 222, 493, 944, 1884, 4020, 8175, 16618, 35198, 73220, 151844, 321036, 676778, 1421828, 3016813, 6407344, 13589888, 28962702, 61853827, 132073646, 282752030, 606492428, 1301587833, 2797816706, 6023460551, 12978238202, 27995493484
Offset: 1

Views

Author

Ilya Gutkovskiy, May 30 2023

Keywords

Crossrefs

Cf. A005750, A007562, A363386.

Programs

  • Mathematica
    nmax = 38; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[A[x^k]^2/k, {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; g[n_] := g[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d g[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 38}]
  • PARI
    seq(n)=my(p=x+x^2+O(x^3)); for(n=1, n\2, my(m=serprec(p,x)-1); p = x + x^2*exp(sum(k=1, m\2, subst(p + O(x^(m\k+1)), x, x^k)^2/k))); Vec(p + O(x*x^n)) \\ Andrew Howroyd, May 30 2023

A363389 G.f. A(x) satisfies: A(x) = x * exp(2 * Sum_{k>=1} A(x^k)^2 / (k*x^k) ).

Original entry on oeis.org

1, 2, 11, 72, 545, 4432, 38081, 339266, 3107841, 29080910, 276786032, 2671136262, 26076724707, 257061506994, 2555287226253, 25584395476368, 257780104545994, 2611791146130284, 26593326491738879, 271972643143865548, 2792566207778712513, 28776796478486084250
Offset: 1

Views

Author

Ilya Gutkovskiy, May 30 2023

Keywords

Crossrefs

Cf. A000151, A005750, A363390.

Programs

  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = x Exp[2 Sum[A[x^k]^2/(k x^k), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; g[n_] := g[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (2/(n - 1)) Sum[Sum[d g[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 22}]
  • PARI
    seq(n)=my(p=x+O(x^2)); for(n=2, n, my(m=serprec(p,x)-1); p = x*exp(2*sum(k=1, m, subst(p + O(x^(m\k+1)), x, x^k)^2/(x^k*k)))); Vec(p) \\ Andrew Howroyd, May 30 2023

A363480 G.f. satisfies A(x) = exp( Sum_{k>=1} A(2*x^k)^2 * x^k/k ).

Original entry on oeis.org

1, 1, 5, 49, 923, 32603, 2198413, 288677317, 74816592016, 38536646525164, 39578607089767640, 81176446754286348780, 332742981886258629407221, 2726830211640382050679262877, 44684572695377447660556579448947
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2023

Keywords

Crossrefs

Cf. A005750, A179470, A363481.

Programs

  • PARI
    seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, subst(A, x, 2*x^k)^2*x^k/k)+x*O(x^n))); Vec(A);

Formula

G.f.: sqrt(B(x)) where B(x) is the g.f. of A363481.

A063689 Number of 2-trees rooted at a triangle with 3 similar edges.

Original entry on oeis.org

1, 1, 2, 6, 21, 83, 356, 1599, 7434, 35381, 171508, 843419, 4197179, 21094355, 106915928, 545859112, 2804656069, 14491370996, 75248398034, 392476363133, 2055245992376, 10801442696736, 56953957110855, 301207378815752, 1597342159296786, 8492297139795170
Offset: 1

Views

Author

Vladeta Jovovic, Aug 22 2001

Keywords

Examples

			Sequence really begins 1, 0, 0, 1, 0, 0, 2, 0, 0, 6, 0, 0, 21, 0, 0, 83, 0, 0, 356, ... but only nonzero trisection is shown.

References

  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 76, Eq. (3.5.17).

Crossrefs

Cf. A005750, A058870, A058866, A054581.

Formula

a(n) = A058870(n) + A063687(n) with A058870(0)=0. - Sean A. Irvine, May 07 2023

Extensions

More terms from Sean A. Irvine, May 07 2023

A063692 Number of 2-trees rooted at a triangle with two similar edges.

Original entry on oeis.org

1, 2, 2, 7, 11, 25, 43, 106, 180, 453, 797, 2023, 3632, 9328, 16960, 44036, 80989, 211815, 393098, 1034958, 1934813, 5121356, 9633260, 25615432, 48433926, 129289382, 245554773, 657691061, 1253974468, 3368475942, 6444250241
Offset: 2

Views

Author

Vladeta Jovovic, Aug 23 2001

Keywords

References

  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 76, Eq. (3.5.16).

Crossrefs

Cf. A005750, A058870, A058866, A054581.
Previous Showing 11-19 of 19 results.

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