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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A249994 Expansion of 1/((1-2*x)*(1+3*x)*(1-4*x)).

Original entry on oeis.org

1, 3, 19, 63, 307, 1095, 4843, 18111, 76483, 294327, 1213147, 4747119, 19308979, 76282599, 308006731, 1223430687, 4919576995, 19600876311, 78636062395, 313847102415, 1257480899731, 5023648225863, 20113423216939, 80397210315903, 321758305696387, 1286524863041655
Offset: 0

Views

Author

Alex Ratushnyak, Dec 28 2014

Keywords

Links

Crossrefs

Cf. A016269 for the expansion of 1/((1-2*x)*(1-3*x)*(1-4*x)).
Cf. A249992, A249993, A249995, A249996.

Programs

  • Magma
    [(5*2^(2*n+3) -7*2^(n+1) +(-1)^n*3^(n+2))/35: n in [0..40]]; // G. C. Greubel, Oct 10 2022
  • Mathematica
    LinearRecurrence[{3,10,-24}, {1,3,19}, 41] (* G. C. Greubel, Oct 10 2022 *)
  • PARI
    Vec(1/((2*x-1)*(3*x+1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Dec 29 2014
  • SageMath
    [(5*2^(2*n+3) -7*2^(n+1) +(-1)^n*3^(n+2))/35 for n in range(41)] # G. C. Greubel, Oct 10 2022

Formula

G.f.: 1/((1-2*x)*(1+3*x)*(1-4*x)).
a(n) = (5*2^(2*n+3) - 7*2^(n+1) + (-1)^n*3^(n+2))/35. - Colin Barker, Dec 29 2014
a(n) = 3*a(n-1) + 10*a(n-2) - 24*a(n-3). - Colin Barker, Dec 29 2014
E.g.f.: (1/35)*(9*exp(-3*x) - 14*exp(2*x) + 40*exp(4*x)). - G. C. Greubel, Oct 10 2022

A249995 Expansion of 1/((1+2*x)*(1-3*x)*(1-4*x)).

Original entry on oeis.org

1, 5, 27, 121, 539, 2289, 9619, 39737, 162987, 663553, 2690051, 10865673, 43783195, 176086097, 707220723, 2837479129, 11375770763, 45580514721, 182554616035, 730915611305, 2925754935291, 11709295114225, 46856010770387, 187480525633401, 750091566966379, 3000874627609409
Offset: 0

Views

Author

Alex Ratushnyak, Dec 28 2014

Keywords

Links

Crossrefs

Cf. A016269 for the expansion of 1/((1-2*x)*(1-3*x)*(1-4*x)).
Cf. A249992, A249993, A249994, A249996.

Programs

  • Magma
    [(5*2^(2*n+3) +(-1)^n*2^(n+1) -3^(n+3))/15: n in [0..40]]; // G. C. Greubel, Oct 10 2022
  • Mathematica
    LinearRecurrence[{5,2,-24}, {1,5,27}, 41] (* G. C. Greubel, Oct 10 2022 *)
CoefficientList[Series[1/((1+2x)(1-3x)(1-4x)),{x,0,40}],x] (* Harvey P. Dale, Oct 28 2022 *)
  • PARI
    Vec(1/((1+2*x)*(1-3*x)*(1-4*x)) + O(x^50)) \\ Michel Marcus, Dec 29 2014
  • SageMath
    [(5*2^(2*n+3) +(-1)^n*2^(n+1) -3^(n+3))/15 for n in range(41)] # G. C. Greubel, Oct 10 2022

Formula

G.f.: 1/((1+2*x)*(1-3*x)*(1-4*x)).
a(n) = ((-1)^n*2^(n+1) + 5*2^(2*n+3) - 3^(n+3))/15. - Colin Barker, Dec 29 2014
a(n) = 5*a(n-1) + 2*a(n-2) - 24*a(n-3). - Colin Barker, Dec 29 2014
E.g.f.: (1/15)*(2*exp(-2*x) - 27*exp(3*x) + 40*exp(4*x)). - G. C. Greubel, Oct 10 2022

A249996 Expansion of 1/((1+2*x)*(1+3*x)*(1-4*x)).

Original entry on oeis.org

1, -1, 15, -5, 191, 99, 2455, 3515, 33231, 74899, 474695, 1371435, 7071871, 23520899, 108399735, 390617755, 1691480111, 6378762099, 26676785575, 103221406475, 423343881951, 1661998662499, 6742129440215, 26686105001595, 107591675061391, 427824901526099, 1718925069371655
Offset: 0

Views

Author

Alex Ratushnyak, Dec 28 2014

Keywords

Links

Crossrefs

Cf. A016269: expansion of 1/((1-2*x)*(1-3*x)*(1-4*x)).
Cf. A249992, A249993, A249994, A249995.

Programs

  • Magma
    [(2^(2*n+3) +(-1)^n*(3^(n+3) -7*2^(n+1)))/21: n in [0..40]]; // G. C. Greubel, Oct 11 2022
  • Mathematica
    LinearRecurrence[{-1,14,24}, {1,-1,15}, 41] (* G. C. Greubel, Oct 11 2022 *)
  • PARI
    Vec(1/((1+2*x)*(1+3*x)*(1-4*x)) + O(x^50)) \\ Michel Marcus, Dec 29 2014
  • SageMath
    [(2^(2*n+3) +(-1)^n*(3^(n+3) -7*2^(n+1)))/21 for n in range(41)] # G. C. Greubel, Oct 11 2022

Formula

G.f.: 1 / ((1+2*x)*(1+3*x)*(1-4*x)).
a(n) = ( 2^(3+2*n) + (3^(3+n)-7*2^(1+n))*(-1)^n )/21. - Colin Barker, Dec 29 2014
a(n) = -a(n-1) + 14*a(n-2) + 24*a(n-3). - Colin Barker, Dec 29 2014
E.g.f.: (1/21)*(27*exp(-3*x) - 14*exp(-2*x) + 8*exp(4*x)). - G. C. Greubel, Oct 11 2022

A084870 Number of 3-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 6, 28, 190, 1692, 16766, 166028, 1586430, 14580412, 129654526, 1123451628, 9544185470, 79881877532, 661135445886, 5425962250828, 44250287565310, 359161631645052, 2904756409742846, 23429320590259628, 188594431902253950
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

Links

Crossrefs

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

Programs

  • Magma
    [(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6: n in [0..50]]; // G. C. Greubel, Oct 08 2017
  • Mathematica
    Table[(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
  • PARI
    for(n=0,50, print1((8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6, ", ")) \\ G. C. Greubel, Oct 08 2017

Formula

a(n) = (1/3!)*(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n).
G.f.: ( 1-26*x+265*x^2-1330*x^3+3340*x^4-3432*x^5 ) / ( (6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(8*x-1)*(5*x-1) ). - R. J. Mathar, Jul 08 2011

A084882 Number of (k,m,n)-multiantichains of multisets with k=3 and m=5.

Original entry on oeis.org

1, 3, 51, 4129, 1439381, 814788851, 395927618035, 155157302244381, 51960586962031617, 15663181302847575559, 4402571746033946222639, 1180812802393866826858193, 306839347397532891662028733
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

Comments

By a (k,m,n)-multiantichain of multisets we mean an m-multiantichain of k-bounded multisets on an n-set. The elements of a multiantichain could have the multiplicities greater than 1. A multiset is called k-bounded if every its element has the multiplicity not greater than k-1.

Links

Crossrefs

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

Programs

  • Mathematica
    Table[(1/5!)*(243^n - 20*162^n + 60*126^n + 20*108^n + 10*98^n - 120*93^n - 120*84^n + 30*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 360*54^n + 720*42^n + 120*36^n - 720*31^n + 275*27^n + 180*26^n - 1650*18^n + 1650*14^n + 870*9^n - 1740*6^n + 744*3^n), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

Formula

a(n) = (1/5!)*(243^n - 20*162^n + 60*126^n + 20*108^n + 10*98^n - 120*93^n - 120*84^n + 30*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 360*54^n + 720*42^n + 120*36^n - 720*31^n + 275*27^n + 180*26^n - 1650*18^n + 1650*14^n + 870*9^n - 1740*6^n + 744*3^n).

A099110 Expansion of 1 / ((1+x)*(1-2x)*(1-3x)*(1-4x)).

Original entry on oeis.org

1, 8, 47, 238, 1113, 4956, 21379, 90266, 375485, 1545544, 6313671, 25650534, 103792417, 418745972, 1685723723, 6775136242, 27197312709, 109079641440, 437189912335, 1751374038590, 7013340021161, 28076893083148
Offset: 0

Views

Author

Ralf Stephan, Sep 28 2004

Keywords

Links

Crossrefs

First differences are in A004057. Pairwise sums are in A016269.

Programs

  • Mathematica
    CoefficientList[Series[1/((1+x)(1-2x)(1-3x)(1-4x)),{x,0,30}],x] (* or *) LinearRecurrence[{8,-17,-2,24},{1,8,47,238},30] (* Harvey P. Dale, Oct 03 2016 *)
  • PARI
    Vec(1/((1+x)*(1-2*x)*(1-3*x)*(1-4*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = (1/60) (80*2^n - 405*3^n + 384*4^n + (-1)^n).
a(n) = 8*a(n-1) - 17*a(n-2) - 2*a(n-3) + 24*a(n-4), n >= 4. - Vincenzo Librandi, Mar 22 2011

A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 8, 19, 9, 1, 0, 16, 65, 55, 14, 1, 0, 32, 211, 285, 125, 20, 1, 0, 64, 665, 1351, 910, 245, 27, 1, 0, 128, 2059, 6069, 5901, 2380, 434, 35, 1, 0, 256, 6305, 26335, 35574, 20181, 5418, 714, 44, 1
Offset: 0

Views

Author

Peter Luschny, Apr 10 2016

Keywords

Examples

			1,
0, 1,
0, 2, 1,
0, 4, 5, 1,
0, 8, 19, 9, 1,
0, 16, 65, 55, 14, 1,
0, 32, 211, 285, 125, 20, 1,
0, 64, 665, 1351, 910, 245, 27, 1.

Links

Crossrefs

Variant: A143494 (the main entry for this triangle).
A005493 (row sums), A074051 (alt. row sums), A000079 (col. 1), A001047 (col. 2),
A016269 (col. 3), A025211 (col. 4), A000096 (diag. n,n-1), A215862 (diag. n,n-2),
A049444, A136124, A143491 (matrix inverse).
Cf. A048993, A269951.

Programs

  • Maple
    A269952 := (n,k) -> Stirling2(n+1, k+1) - Stirling2(n, k+1):
seq(seq(A269952(n,k), k=0..n), n=0..9);
  • Mathematica
    Flatten[ Table[ Sum[(-1)^(n-j) Binomial[-j,-n] StirlingS2[j,k], {j,0,n}], {n,0,9}, {k,0,n}]]

Formula

T(n, k) = S2(n+1, k+1) - S2(n, k+1).

A381067 Expansion of e.g.f. log(1-x)^2 * exp(-x) / (2 * (1-x)).

Original entry on oeis.org

0, 0, 1, 3, 17, 100, 694, 5453, 48082, 470328, 5057331, 59313287, 753695139, 10316991100, 151373235896, 2370151632977, 39450142911652, 695612154233648, 12953591498092101, 254044853932550091, 5234026736314790581, 113025076301648693844, 2552830193825115461786
Offset: 0

Views

Author

Seiichi Manyama, Feb 12 2025

Keywords

Crossrefs

Column k=3 of A269954 (with a different offset).
Cf. A016269, A381024, A381065.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*abs(stirling(k+1, 3, 1)));

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * |Stirling1(k+1,3)|.
a(n) = A381065(n) + A381065(n+1).

A032871 Numbers whose base-8 representation Sum_{i=0..m} d(i)*8^i has d(m) >= d(m-1) <= d(m-2) >= ...

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18, 24, 25, 26, 27, 32, 33, 34, 35, 36, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 128, 129, 130, 131, 132
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A016269.

A059090 Triangle T(n,m) giving number of m-element intersecting antichains on a labeled n-set or n-variable Boolean functions with m nonzero values in the Post class F(7,2), m=0,.., A037952(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 7, 3, 1, 1, 15, 30, 30, 5, 1, 31, 195, 605, 780, 543, 300, 135, 45, 10, 1, 1, 63, 1050, 9030, 41545, 118629, 233821, 329205, 327915, 224280, 100716, 29337, 5950, 910, 105, 7
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Dec 28 2000

Keywords

Comments

An antichain is called intersecting (or proper) antichain if every two members have a nonempty intersection. Row sums give the number of intersecting antichains on a labeled n-set or n-variable Boolean functions in the Post class F(7,2) or self-dual monotone Boolean functions of n+1 variables. Cf. A001206.

Examples

			1;
1, 1;
1, 3;
1, 7, 3, 1;
1, 15, 30, 30, 5;
1, 31, 195, 605, 780, 543, 300, 135, 45, 10, 1;
1, 63, 1050, 9030, 41545, 118629, 233821, 329205, 327915, 224280, 100716, 29337, 5950, 910, 105, 7;

References

  • Jovovic V., Kilibarda G., The number of n-variable Boolean functions in the Post class F(7,2), Belgrade, 2001, in preparation.
  • Pogosyan G., Miyakawa M., Nozaki A., Rosenberg I., The Number of Clique Boolean Functions, IEICE Trans. Fundamentals, Vol. E80-A, No. 8, pp. 1502-1507, 1997/8.

Links

Crossrefs

Cf. A001206, A032263, A051303-A051307, A036239, A051180-A051185, A016269, A047707, A051112-A051118, A000372.

Formula

T(n, 0)=1, T(n, 1)=2^n-1, T(n, 2)=A032263(n), T(n, 3)=A051303(n), T(n, 4)=A051304(n), T(n, 5)=A051305(n), T(n, 6)=A051306(n), T(n, 7)=A051307(n).
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