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.

A308946 Expansion of e.g.f. 1/(1 - x*(1 + x/2)*exp(x)).

Original entry on oeis.org

1, 1, 5, 30, 244, 2485, 30351, 432502, 7043660, 129050649, 2627117875, 58829021416, 1437117395946, 38032508860177, 1083932872119839, 33098858988564090, 1078083456543449416, 37309607437056658129, 1367138649165397662627, 52879280631976735387588
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 02 2019

Keywords

Crossrefs

Cf. A000217, A001793, A006153, A052529, A279361, A308861.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 - x (1 + x/2) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k + 1, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

Formula

E.g.f.: 1 / (1 - Sum_{k>=1} (k*(k + 1)/2)*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000217(k) * a(n-k).
a(n) ~ n! * (2 + r) / ((2 + 4*r + r^2) * r^n), where r = 0.49122518354447387971550543450091640839121607... is the root of the equation exp(r)*r*(2 + r) = 2. - Vaclav Kotesovec, Aug 09 2021

A300455 Logarithmic transform of the triangular numbers A000217.

Original entry on oeis.org

0, 1, 2, -1, -11, 19, 201, -764, -7426, 52137, 448435, -5377604, -38712486, 777663613, 4258812299, -149524753650, -505685566184, 36733876797025, 30910872539763, -11174584391207360, 25170998506744790, 4101787001153848461, -24862093152821214653, -1776483826032814964966
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 06 2018

Keywords

Examples

			E.g.f.: A(x) = x/1! + 2*x^2/2! - x^3/3! - 11*x^4/4! + 19*x^5/5! + 201*x^6/6! - 764*x^7/7! - 7426*x^8/8! + ...

Links

Crossrefs

Cf. A000217, A009306, A033464, A279361, A300452.

Programs

  • Maple
    a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n) -add(j*
      binomial(n, j)*t(n-j)*a(j), j=1..n-1)/n))(i->i*(i+1)/2)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 06 2018
  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 + Exp[x] x (x + 2)/2], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: log(1 + exp(x)*x*(x + 2)/2).

A281231 Exponential transform of the tetrahedral numbers (A000292).

Original entry on oeis.org

1, 1, 5, 23, 133, 916, 7107, 61286, 580505, 5968400, 66032901, 780962524, 9817927385, 130572957724, 1829676460991, 26919714974436, 414591408939313, 6665930432840304, 111624874150941193, 1942675652654112012, 35071252458352443001, 655641049733709757516
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 18 2017

Keywords

Examples

			E.g.f.: A(x) = 1 + x/1! + 5*x^2/2! + 23*x^3/3! + 133*x^4/4! + 916*x^5/5! + 7107*x^6/6! + ...

Links

Crossrefs

Cf. A000292, A279361, A279358.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*j*(j+1)*(j+2)/6, j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 18 2017
  • Mathematica
    Range[0, 21]! CoefficientList[Series[Exp[Exp[x] x (1 + x + x^2/6)], {x, 0, 21}], x]

Formula

E.g.f.: exp(exp(x)*x*(1+x+x^2/6)).

A320254 a(n) = n! * [x^n] exp(exp(x)*(x + (n/2 - 1)*x^2)).

Original entry on oeis.org

1, 1, 3, 16, 125, 1291, 16177, 241207, 4153193, 81082225, 1770989921, 42763506919, 1131353484637, 32541516492811, 1011058416700529, 33745374949198231, 1204107124715441873, 45741398365345761073, 1843069565594762478145, 78511973999963036415967, 3525468554804288803649381
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 08 2018

Keywords

Comments

For n > 2, a(n) is the n-th term of the exponential transform of n-gonal numbers.

Links

Crossrefs

Cf. A000248, A033462, A279361, A279644, A318118, A318124, A320255.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[Exp[x] (x + (n/2 - 1) x^2)], {x, 0, n}], {n, 0, 20}]

A336961 Expansion of e.g.f. exp(x * (2 + x) * exp(x)).

Original entry on oeis.org

1, 2, 10, 56, 384, 3022, 26626, 258624, 2734360, 31168682, 380196414, 4932536908, 67717987948, 979613124414, 14877703575130, 236469561581768, 3922587278751504, 67743812585483218, 1215417753459838198, 22609895367286957572, 435341977596130683316
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 09 2020

Keywords

Comments

Exponential transform of the oblong numbers (A002378).

Crossrefs

Cf. A000248, A002378, A033462, A209801, A216507, A279361, A336960.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[x (2 + x) Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] k (k + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

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

A347665 E.g.f.: exp( exp(x) * (1 + x + x^2 / 2) - 1 ).

Original entry on oeis.org

1, 2, 8, 39, 227, 1518, 11368, 93796, 842416, 8158942, 84581560, 932878169, 10891741957, 134043979644, 1732583270218, 23445954950207, 331260511278659, 4874617929283392, 74548457001207068, 1182551615010825076, 19423368875596930596, 329809489306236629874
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2021

Keywords

Comments

Exponential transform of A000124.

Crossrefs

Cf. A000085, A000124, A209801, A279361, A347666.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[Exp[x] (1 + x + x^2/2) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (k (k + 1)/2 + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A000124(k) * a(n-k).

A294221 Exponential transform of the square pyramidal numbers (A000330).

Original entry on oeis.org

1, 1, 6, 30, 192, 1471, 12637, 120723, 1267492, 14438913, 176961001, 2318180239, 32275104644, 475285152707, 7373223596299, 120078748361611, 2046720320727328, 36414341169682417, 674650306604656821, 12988470845576660407, 259348785562811740236, 5361803880323803698731, 114593610390850499426211
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 25 2017

Keywords

Examples

			E.g.f.: A(x) = 1 + x/1! + 6*x^2/2! + 30*x^3/3! + 192*x^4/4! + 1471*x^5/5! + 12637*x^6/6! + ...

Links

Crossrefs

Cf. A000330, A033462, A279361, A281231.

Programs

  • Mathematica
    Range[0, 22]! CoefficientList[Series[Exp[Exp[x] x (6 + 9 x + 2 x^2)/6], {x, 0, 22}], x]
a[n_] := a[n] = Sum[a[n - k] Binomial[n - 1, k - 1] k (k + 1) (2 k + 1)/6, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: exp(exp(x)*x*(6 + 9*x + 2*x^2)/6).

A352658 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * binomial(k+1,2) * k * a(n-k).

Original entry on oeis.org

1, 1, 5, 39, 508, 9235, 224481, 6959932, 266492388, 12302514945, 671505310855, 42664357009186, 3114726872133570, 258452373177094213, 24149855477595375815, 2520813303733886387220, 291892618561012451083816, 37264133443594227118861233, 5216461719269145457350349359
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2022

Keywords

Crossrefs

Cf. A000217, A002411, A023998, A279361, A336227, A337591.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 Binomial[k + 1, 2] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(x BesselI[0, 2 Sqrt[x]] + Sqrt[x] BesselI[1, 2 Sqrt[x]])/2], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / n!^2 = exp( (x * BesselI(0,2*sqrt(x)) + sqrt(x) * BesselI(1,2*sqrt(x))) / 2 ).
Sum_{n>=0} a(n) * x^n / n!^2 = exp( Sum_{n>=1} binomial(n+1,2) * x^n / n!^2 ).

A372623 Expansion of e.g.f. exp( exp(x) * (1 + x^2 / 2) - 1 ).

Original entry on oeis.org

1, 1, 3, 11, 48, 247, 1448, 9445, 67651, 526704, 4418875, 39670270, 378931567, 3832882393, 40886570975, 458341921775, 5382862509572, 66050096110691, 844741961321026, 11236481306649167, 155150031880549077, 2219877203279634396, 32860282502526114729
Offset: 0

Views

Author

Ilya Gutkovskiy, May 07 2024

Keywords

Crossrefs

Cf. A000124, A209801, A279361.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Exp[x] (1 + x^2/2) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (k (k - 1)/2 + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A000124(k-1) * a(n-k).
Showing 1-9 of 9 results.

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