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.

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A111673 Triangle, generated from A111579.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 15, 11, 4, 1, 1, 1, 52, 49, 19, 5, 1, 1, 1, 203, 257, 109, 29, 6, 1, 1, 1, 877, 1539, 742, 201, 41, 7, 1, 1, 1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1, 1, 21147, 75905, 51193, 15821, 3176, 505, 71, 9, 1, 1, 1, 115975, 609441, 498118, 170389, 35451, 5497, 729, 89, 10, 1, 1
Offset: 0

Views

Author

Gary W. Adamson, Aug 14 2005

Keywords

Comments

Columns are inverse binomial transforms of columns (k>0) of A111579.

Examples

			First few rows of the triangle are:
  1,
  1, 1,
  1, 1, 1,
  1, 2, 1, 1,
  1, 5, 3, 1, 1,
  1, 15, 11, 4, 1, 1,
  1, 52, 49, 19, 5, 1, 1,
  1, 203, 257, 109, 29, 6, 1, 1,
  1, 877, 1539, 742, 201, 41, 7, 1, 1,
  1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1,
  ...
Inverse binomial transform of column 2 of A111579 (1, 2, 5, 15, 52, 203...) = column 2 (1, 1, 2, 5, 15, 52...).

Links

Crossrefs

Cf. A111579, A008277.
For two other versions of this triangle see A241578, A241579.
Rows and columns give A000110, A004211, A004212, A004213, A005011, A005012, A028387, A241577.

Extensions

More terms from N. J. A. Sloane, Apr 29 2014

A119430 Expansion of Sum_{k>=0} 2^k*x^(2k)/Product_{j=1..k} (1 - j*2x).

Original entry on oeis.org

1, 0, 2, 4, 12, 40, 152, 640, 2928, 14400, 75744, 424640, 2527552, 15902848, 105313408, 731376640, 5311088896, 40233525248, 317296341504, 2600091120640, 22099119279104, 194487001540608, 1769555559897088, 16622286300921856
Offset: 0

Views

Author

Paul Barry, May 19 2006

Keywords

Links

Crossrefs

Cf. A004211, A048993, A119429.

Programs

  • Mathematica
    a[n_] := Sum[2^(n-k) * StirlingS2[n - k, k], {k, 0, Floor[n/2]}]; Array[a, 25, 0] (* Amiram Eldar, Apr 09 2022 *)
  • PARI
    a(n) = sum(k=0, n\2, 2^(n-k)*stirling(n-k, k, 2)); \\ Seiichi Manyama, Apr 08 2022

Formula

a(n) = Sum_{k=0..n} S2(k,n-k)*2^k where S2(n,k)=A048993(n,k);
a(n) = Sum_{k=0..floor(n/2)} S2(n-k,k)*2^(n-k).

A308543 Expansion of e.g.f. exp(2*(exp(2*x) - 1)).

Original entry on oeis.org

1, 4, 24, 176, 1504, 14528, 155520, 1819392, 23019008, 312413184, 4518705152, 69279690752, 1120856170496, 19062628335616, 339681346551808, 6323658075340800, 122680376836358144, 2474677219852288000, 51799971194270646272, 1123121391647711035392
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 06 2019

Keywords

Crossrefs

Cf. A000079, A000110, A001861, A002866, A004211, A055882.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[2 (Exp[2 x] - 1)], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Sum[4^k x^k/Product[(1 - 2 j x), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = Sum[2^(k + 1) Binomial[n - 1, k - 1] a[n - k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]
Table[2^n BellB[n, 2], {n, 0, 19}]

Formula

O.g.f.: Sum_{k>=0} 4^k*x^k / Product_{j=1..k} (1 - 2*j*x).
E.g.f.: exp(4*exp(x)*sinh(x)).
E.g.f.: g(g(x) - 1), where g(x) = e.g.f. of A000079 (powers of 2).
E.g.f.: f(x)^4, where f(x) = e.g.f. of A004211 (shifts one place left under 2nd-order binomial transform).
a(0) = 1; a(n) = Sum_{k=1..n} 2^(k+1)*binomial(n-1,k-1)*a(n-k).
a(n) = Sum_{k=0..n} 2^(n+k)*Stirling2(n,k).
a(n) = exp(-2) * Sum_{k>=0} 2^(n+k)*k^n/k!.
a(n) = 2^n * A001861(n).

A337010 a(n) = exp(-1/2) * Sum_{k>=0} (2*k + 3)^n / (2^k * k!).

Original entry on oeis.org

1, 4, 18, 92, 532, 3440, 24552, 191280, 1612304, 14597952, 141123872, 1449324992, 15743376704, 180203389696, 2166381979264, 27274611880704, 358690234163456, 4916123783848960, 70076765972288000, 1036967662211324928, 15902394743591408640
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2020

Keywords

Crossrefs

Cf. A004211, A005494, A007405, A337011.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[3 x + (Exp[2 x] - 1)/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 4 a[n - 1] + Sum[Binomial[n - 1, k - 1] 2^(k - 1) a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 20}]

Formula

E.g.f.: exp(3*x + (exp(2*x) - 1) / 2).
a(0) = 1; a(n) = 4 * a(n-1) + Sum_{k=2..n} binomial(n-1,k-1) * 2^(k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * A004211(k+1).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * A004211(k).

A367743 Expansion of e.g.f. exp(1 - x - exp(2*x)).

Original entry on oeis.org

1, -3, 5, 1, -7, -75, -99, 1241, 10161, 18989, -332299, -3857551, -14440151, 141168997, 2807256333, 20182451657, -42073176479, -2999363709091, -38439478980891, -161835672017439, 3439471815545177, 87228227501354517, 937579822282327421, 216540362854403513, -198501712690150659055
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 29 2023

Keywords

Crossrefs

Cf. A000587, A004211, A009235, A109747, A124311, A126390, A308536, A308645, A367744.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[1 - x - Exp[2 x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] - Sum[Binomial[n - 1, k - 1] 2^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k BellB[k, -1], {k, 0, n}], {n, 0, 24}]

Formula

a(n) = exp(1) * Sum_{k>=0} (-1)^k * (2*k-1)^n / k!.
a(0) = 1; a(n) = -a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * 2^k * a(n-k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * A000587(k).

A380257 Expansion of e.g.f. exp( (1/(1-3*x)^(2/3) - 1)/2 ).

Original entry on oeis.org

1, 1, 6, 56, 706, 11186, 213156, 4742256, 120571676, 3447128796, 109427729096, 3818008773536, 145196289453656, 5976489668054296, 264685744187399536, 12548508890339297856, 634022724191046592016, 34007862777419093053456, 1929842567333195106456416
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2025

Keywords

Links

Crossrefs

Cf. A049376, A375173, A380258.
Cf. A004211, A380214.

Programs

  • Mathematica
    CoefficientList[Series[Exp[ (1/(1-3*x)^(2/3) - 1)/2 ],{x,0,18}],x]Range[0,18]! (* Stefano Spezia, Mar 31 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-3*x)^(2/3)-1)/2)))

Formula

a(n) = Sum_{k=0..n} 3^(n-k) * |Stirling1(n,k)| * A004211(k) = Sum_{k=0..n} 2^k * 3^(n-k) * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.
a(n) = (1/exp(1/2)) * (-3)^n * n! * Sum_{k>=0} binomial(-2*k/3,n)/(2^k * k!).

A380261 Expansion of e.g.f. exp( ((1+3*x)^(2/3) - 1)/2 ).

Original entry on oeis.org

1, 1, 0, 2, -14, 146, -1944, 31620, -608068, 13502076, -340052704, 9579145016, -298455813160, 10191129869272, -378469678855904, 15187759126892976, -654936026064200944, 30203464484648818960, -1483333523694819075328, 77291514214052885054496
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2025

Keywords

Links

Crossrefs

Cf. A000085, A002119, A004211, A380262.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(((1+3*x)^(2/3)-1)/2)))

Formula

a(n) = Sum_{k=0..n} 3^(n-k) * Stirling1(n,k) * A004211(k) = Sum_{k=0..n} 2^k * 3^(n-k) * Stirling1(n,k) * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.
a(n) = (1/exp(1/2)) * 3^n * n! * Sum_{k>=0} binomial(2*k/3,n)/(2^k * k!).

A383206 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = Sum_{j=k..n} 2^(n-j) * Stirling2(n,j) * Stirling2(j,k).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 11, 9, 1, 0, 49, 71, 18, 1, 0, 257, 575, 245, 30, 1, 0, 1539, 4957, 3120, 625, 45, 1, 0, 10299, 45829, 39697, 11480, 1330, 63, 1, 0, 75905, 454015, 517790, 201677, 33250, 2506, 84, 1, 0, 609441, 4804191, 6999785, 3513762, 770007, 81774, 4326, 108, 1
Offset: 0

Views

Author

Seiichi Manyama, Apr 19 2025

Keywords

Examples

			Triangle starts:
  1;
  0,     1;
  0,     3,     1;
  0,    11,     9,     1;
  0,    49,    71,    18,     1;
  0,   257,   575,   245,    30,    1;
  0,  1539,  4957,  3120,   625,   45,  1;
  0, 10299, 45829, 39697, 11480, 1330, 63, 1;
  ...

Crossrefs

Columns k=0..3 give A000007, A004211 (for n > 0), A383207, A383208.
Row sums give A380228.
Cf. A130191.

Programs

  • PARI
    T(n, k) = sum(j=k, n, 2^(n-j)*stirling(n, j, 2)*stirling(j, k, 2));

Formula

E.g.f. of column k (with leading zeros): (exp(f(x)) - 1)^k / k! with f(x) = (exp(2*x) - 1)/2.

A186001 Expansion of 1/(1-x/(1-x/(1-x/(1-2x/(1-x/(1-4x/(1-x/(1-6x/(1-... (continued fraction).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 205, 917, 4658, 26817, 173899, 1257916, 10036409, 87351977, 821268402, 8273013085, 88696718215, 1006586902196, 12037812901733, 151136735414301, 1985815346287906
Offset: 0

Views

Author

Paul Barry, Feb 09 2011

Keywords

Comments

Hankel transform is A186002.

Crossrefs

Cf. A004211, A186002.

Formula

G.f.: 1/(1-x/(1-x*g(x))), g(x) the g.f. of A004211.
G.f.: 1/(1-x-x^2/(1-2x-2x^2/(1-3x-4x^2/(1-5x-6x^2/(1-7x-8x^2/(1-.... (continued fraction).
a(n) = a(n-1) + Sum_{i=0..n-2} A004211(i)*a(n-1-i) for n > 0.

A305708 Expansion of e.g.f. exp(cos(x)/exp(x) - 1).

Original entry on oeis.org

1, -1, 1, 1, -11, 43, -83, -275, 3833, -21561, 51369, 375593, -5860147, 40452371, -101676235, -1409619211, 23912208945, -189650997937, 454996127889, 11250036170129, -204691511497499, 1799897065507003, -3741969787709699, -164548323889940675, 3183842522596250537, -30356999697044585833
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 08 2018

Keywords

Examples

			exp(cos(x)/exp(x) - 1) = 1 - x + x^2/2! + x^3/3! - 11*x^4/4! + 43*x^5/5! - 83*x^6/6! - 275*x^7/7! + ...

Crossrefs

Cf. A004211, A009116, A009216, A009235, A009255, A009276, A146559, A155585.

Programs

  • Maple
    a:=series(exp(cos(x)/exp(x)-1),x=0,26): seq(n!*coeff(a,x,n),n=0..25); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Cos[x]/Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Re[(-1 - I)^k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 25}]
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