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 17 results. Next

A238214 Decimal expansion of a constant related to A143917.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Feb 21 2014

Keywords

Examples

			1.81857005675331400362707139219522893237...

Crossrefs

Cf. A143917, A143916.

Formula

Equals lim n->infinity A143917(n)/n!.

A385843 a(n) = 1 + Sum_{k=0..n-1} k^5 * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 66, 16105, 16507753, 51603272051, 401318681776723, 6745364508844808841, 221038850400001766938953, 13052344129663319516736911260, 1305247465753403752473945799113276, 210212714880649951675343095297590137757, 52307860484508916277278208388919504757392477
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A143917, A385840, A385841, A385842.
Cf. A385833, A385838.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, j^5*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*Sum_{k=1..5} Stirling2(5,k) * x^k * (d^k/dx^k A(x)) ) ).

A385874 a(n) = 1 + Sum_{k=0..n-1} binomial(k+1,2) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 8, 57, 639, 10357, 229588, 6686619, 248013315, 11425386222, 640413284553, 42933889931191, 3393203732253145, 312268381507616935, 33107736233111305459, 4006699123399932333697, 548987463226205098599755, 84552444466155546810368421, 14544161652321384236939516147
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2025

Keywords

Links

Crossrefs

Cf. A143917, A385875, A385876, A385877.
Cf. A385830, A385835, A385840.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, binomial(j+1, 2)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x^2 * (d/dx A(x)) - x^3/2 * (d^2/dx^2 A(x)) ) ).

A143916 G.f. A(x) satisfies: A(x) = 1+x + x^2*A(x)*A'(x).

Original entry on oeis.org

1, 1, 1, 3, 12, 62, 385, 2781, 22848, 210176, 2139336, 23872450, 289825228, 3803859030, 53676793157, 810508456373, 13041332257860, 222776899815744, 4026846590787586, 76792054455516582, 1540845309830989064
Offset: 0

Views

Author

Paul D. Hanna, Sep 05 2008

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 12*x^4 + 62*x^5 + 385*x^6 +...
A'(x) = 1 + 2*x + 9*x^2 + 48*x^3 + 310*x^4 + 2310*x^5 + 19467*x^6 +...
A(x)*A'(x) = 1 + 3*x + 12*x^2 + 62*x^3 + 385*x^4 + 2781*x^5 +...

Links

Crossrefs

Cf. A143917 (variant), A238214.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[n_] := a[n] = Sum[k a[k] a[n-k-1], {k, 1, n-1}]; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, May 17 2016 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x+x^2*deriv(A^2/2)); polcoeff(A, n)}

Formula

a(n) ~ c * n!, where constant c = A238214 / exp(1) = 0.669014536209527303065690569951975534726... - Vaclav Kotesovec, Feb 21 2014
a(0) = 1, a(1) = 1, a(n) = Sum_{0 < k < n} k * a(k) * a(n-k-1). - Vladimir Reshetnikov, May 17 2016

A385840 a(n) = 1 + Sum_{k=0..n-1} k^2 * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 10, 101, 1733, 45303, 1680907, 84166419, 5475072843, 449157456364, 45377436182152, 5537042709272831, 802969519178558759, 136516626968319610486, 26895468447194766859402, 6078661245454015521843883, 1562271796018872884111521763, 453071380100390505646644605866
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Links

Crossrefs

Cf. A143917, A385841, A385842, A385843.
Cf. A385830, A385835, A385874.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, j^2*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x^2 * (d/dx A(x)) - x^3 * (d^2/dx^2 A(x)) ) ).

A385841 a(n) = 1 + Sum_{k=0..n-1} k^3 * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 18, 505, 32857, 4141211, 898723027, 309170208201, 158606268801081, 115783226426053396, 115899337245305115516, 154378153899481307826141, 266920063540268509322880013, 586690612016923635703423527652, 1610466268575965949949881680290412
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A143917, A385840, A385842, A385843.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, j^3*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*Sum_{k=1..3} Stirling2(3,k) * x^k * (d^k/dx^k A(x)) ) ).

A385842 a(n) = 1 + Sum_{k=0..n-1} k^4 * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 34, 2789, 716837, 448746495, 582025808335, 1398026940957747, 5727717572863611987, 37585285548218779674700, 375890452313654055440508988, 5503788078310849677217561978523, 114132054134076966886682122559148347, 3259839741208602005078393364829175139526
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A143917, A385840, A385841, A385843.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, j^4*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*Sum_{k=1..4} Stirling2(4,k) * x^k * (d^k/dx^k A(x)) ) ).

A385875 a(n) = 1 + Sum_{k=0..n-1} binomial(k+2,3) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 10, 111, 2347, 84757, 4837213, 411373408, 49787445476, 8265626303452, 1826809978098228, 524311794034090050, 191377585766768936606, 87269255118865044728501, 48958442598180565027265909, 33340876732769115354996751746, 27239595466972699678481509900786
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2025

Keywords

Crossrefs

Cf. A143917, A385874, A385876, A385877.
Cf. A385841.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, binomial(j+2, 3)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*Sum_{k=1..3} binomial(2,k-1) * x^k/k! * (d^k/dx^k A(x)) ) ).

A385876 a(n) = 1 + Sum_{k=0..n-1} binomial(k+3,4) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 12, 193, 6968, 495189, 62906143, 13274340034, 4393943557987, 2179423896462618, 1560476564415661780, 1563601961040080858376, 2135883440687340361131857, 3889446901597262416621276499, 9260777373178278371280728311304, 28347247357191779349093896687278933
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2025

Keywords

Crossrefs

Cf. A143917, A385874, A385875, A385877.
Cf. A385842.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, binomial(j+3, 4)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*Sum_{k=1..4} binomial(3,k-1) * x^k/k! * (d^k/dx^k A(x)) ) ).

A385877 a(n) = 1 + Sum_{k=0..n-1} binomial(k+4,5) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 14, 309, 17637, 2240632, 566921596, 262489646519, 208155482551991, 268104800528280951, 537014337938584568385, 1613191612128443060280697, 7048035233444754041436840277, 43620293298146615746333469478901, 373782307403691698916363133787269075
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2025

Keywords

Crossrefs

Cf. A143917, A385874, A385875, A385876.
Cf. A385843.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, binomial(j+4, 5)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*Sum_{k=1..5} binomial(4,k-1) * x^k/k! * (d^k/dx^k A(x)) ) ).
Showing 1-10 of 17 results. Next

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