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 10 results.

A246312 Decimal expansion of a constant related to identity matched trees.

Original entry on oeis.org

5, 2, 4, 9, 0, 3, 2, 4, 9, 1, 2, 2, 8, 1, 7, 0, 5, 7, 9, 1, 6, 4, 9, 5, 2, 2, 1, 6, 1, 8, 4, 3, 0, 9, 2, 6, 5, 3, 4, 3, 0, 8, 6, 3, 3, 7, 6, 4, 8, 7, 3, 6, 5, 0, 3, 2, 0, 2, 2, 3, 3, 1, 8, 6, 0, 5, 9, 5, 8, 5, 5, 6, 5, 2, 6, 4, 0, 2, 8, 7, 7, 5, 8, 7, 0, 4, 5, 7, 4, 4, 0, 9, 9, 4, 5, 1, 8, 6, 5, 4, 7, 3, 8, 7, 6
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 25 2014

Keywords

Examples

			5.249032491228170579164952216184309265343086337648736503202233186059585565...

Crossrefs

Cf. A005753, A005754, A005755, A102755, A038078, A038077.

Formula

Equals lim n -> infinity A005753(n)^(1/n).
Equals lim n -> infinity A005754(n)^(1/n).
Equals lim n -> infinity A005755(n)^(1/n).
Equals lim n -> infinity A102755(n)^(1/n).
Equals lim n -> infinity A038078(n)^(1/n).

Extensions

More terms from Vaclav Kotesovec, Sep 06 2014, Feb 24 2015 and Dec 26 2020

A052755 G.f. A(x) satisfies: A(x) = exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^3 * x^k / k ).

Original entry on oeis.org

1, 1, 3, 15, 79, 466, 2872, 18409, 121197, 815491, 5581214, 38737651, 272012178, 1928939678, 13794498614, 99371002295, 720411445866, 5252194141946, 38482834469488, 283223825607253, 2092829973445703
Offset: 0

Views

Author

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

Keywords

Comments

Old name was: A simple grammar.

Links

Crossrefs

Cf. A005754, A052775, A052798.

Programs

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

Formula

G.f. A(x) satisfies: A(x) = exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^3 * x^k / k ). - Ilya Gutkovskiy, May 26 2023

Extensions

New name from Ilya Gutkovskiy, May 26 2023

A052775 G.f. A(x) satisfies: A(x) = exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^4 * x^k / k ).

Original entry on oeis.org

1, 1, 4, 26, 184, 1443, 11888, 101859, 897529, 8085103, 74113656, 689134849, 6484074328, 61620879930, 590628242876, 5703027934533, 55423681958153, 541689157201498, 5320989368024126, 52503593913927276
Offset: 0

Views

Author

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

Keywords

Comments

Old name was: A simple grammar.

Links

Crossrefs

Cf. A005754, A052755, A052798.

Programs

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

Formula

G.f. A(x) satisfies: A(x) = exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^4 * x^k / k ). - Ilya Gutkovskiy, May 26 2023

Extensions

New name from Ilya Gutkovskiy, May 26 2023

A363388 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^2 / (k*x^k) ).

Original entry on oeis.org

1, 1, 1, 2, 5, 10, 28, 70, 190, 517, 1441, 4057, 11572, 33294, 96620, 282319, 830178, 2454384, 7292106, 21759413, 65185967, 195976025, 591097127, 1788122219, 5423917828, 16493458475, 50270190728, 153544874713, 469916030995, 1440807810639, 4425266768759, 13613578089594, 41943137192265
Offset: 1

Views

Author

Ilya Gutkovskiy, May 30 2023

Keywords

Crossrefs

Cf. A005754, A007560, A363387.

Programs

  • Mathematica
    nmax = 33; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^(k + 1) A[x^k]^2/(k x^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[(-1)^(k/d + 1) d g[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 33}]
  • 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, (-1)^k*subst(p + O(x^(m\k+1)), x, x^k)^2/(x^k*k)))); Vec(p + O(x*x^n)) \\ Andrew Howroyd, May 30 2023

A005755 Number of identity matched trees with n nodes.

Original entry on oeis.org

0, 0, 0, 1, 4, 16, 64, 252, 1018, 4182, 17510, 74510, 322034, 1410362, 6251114, 27998532, 126583634, 577079333, 2650573354, 12256481666, 57021299394, 266754944481, 1254245360430, 5924659521632, 28105641930102, 133853504339029, 639801068848128
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A005753, A005754, A102755, A246312.

Programs

  • Maple
    with(numtheory): b2:= proc(n) option remember; local m; `if`(n=1, 1, 2/(n-1) *add(b2(m) *add((-1)^((n-m)/d+1) *d*b2(d), d=divisors(n-m)), m=1..n-1)) end: c2:= proc(n) option remember; local m; `if`(n=1, 1, 1/(n-1) *add(c2(m) *add((-1)^((n-m)/d+1) *d*b2(d), d=divisors(n-m)), m=1..n-1)) end: a2:= n-> (b2(n) -add(b2(m) *b2(n-m), m=1..n-1) -`if`(irem(n, 2)=0, b2(n/2), c2((n+1)/2)))/2: seq(a2(n), n=1..30); # Alois P. Heinz, Aug 04 2009
  • Mathematica
    b2[n_] := b2[n] = If [n == 1, 1, 2/(n-1)*Sum[b2[m]*Sum[(-1)^((n-m)/d+1)*d*b2[d], {d, Divisors[n-m]}], {m, 1, n-1}]]; c2[n_] := c2[n] = If [n == 1, 1, 1/(n-1)*Sum[c2[m]*Sum[(-1)^((n-m)/d+1)*d*b2[d], {d, Divisors[n-m]}], {m, 1, n-1}]]; a2[n_] := (b2[n] - Sum[b2[m]*b2[n-m], {m, 1, n-1}] - If[Mod[n, 2] == 0, b2[n/2], c2[(n+1)/2]])/2; Table[a2[n], {n, 1, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / n^(5/2), where d = A246312 = 5.2490324912281705791649522..., c = 0.089035519570392123219315... . - Vaclav Kotesovec, Aug 25 2014

Extensions

More terms from Alois P. Heinz, Aug 04 2009

A052798 G.f. A(x) satisfies: A(x) = exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^5 * x^k / k ).

Original entry on oeis.org

1, 1, 5, 40, 355, 3475, 35836, 384436, 4243860, 47905385, 550404336, 6415528666, 75677788275, 901728156490, 10837196405920, 131215506276862, 1599078373019073, 19598996116313001, 241433496694878595
Offset: 0

Views

Author

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

Keywords

Comments

Old name was: A simple grammar.

Links

Crossrefs

Cf. A005754, A052755, A052775.

Programs

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

Formula

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

Extensions

New name from Ilya Gutkovskiy, May 26 2023

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

A363386 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^2 / k ).

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 4, 8, 10, 22, 50, 77, 160, 343, 622, 1250, 2648, 5127, 10364, 21685, 43594, 88907, 185458, 380113, 782902, 1633841, 3387444, 7033401, 14716304, 30734066, 64228198, 134824862, 283040684, 594516622, 1252151812, 2639220817, 5566237724, 11760037378
Offset: 1

Views

Author

Ilya Gutkovskiy, May 30 2023

Keywords

Crossrefs

Cf. A005754, A007560, A363385.

Programs

  • Mathematica
    nmax = 38; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^(k + 1) 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[(-1)^(k/d + 1) 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, (-1)^k*subst(p + O(x^(m\k+1)), x, x^k)^2/k))); Vec(p + O(x*x^n)) \\ Andrew Howroyd, May 30 2023

A363390 G.f. A(x) satisfies: A(x) = x * exp(2 * Sum_{k>=1} (-1)^(k+1) * A(x^k)^2 / (k*x^k) ).

Original entry on oeis.org

1, 2, 9, 60, 436, 3462, 28810, 248606, 2202772, 19929336, 183331451, 1709642222, 16125333248, 153564283602, 1474528190435, 14260019116712, 138772479615509, 1357948477513772, 13353454737592303, 131889469476063586, 1307802326452419584, 13014461023695752740
Offset: 1

Views

Author

Ilya Gutkovskiy, May 30 2023

Keywords

Crossrefs

Cf. A005753, A005754, A363389.

Programs

  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = x Exp[2 Sum[(-1)^(k + 1) 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[(-1)^(k/d + 1) 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, (-1)^k*subst(p + O(x^(m\k+1)), x, x^k)^2/(x^k*k)))); Vec(p) \\ Andrew Howroyd, May 30 2023
Showing 1-10 of 10 results.

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