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 1-4 of 4 results.

A016269 Number of monotone Boolean functions of n variables with 2 mincuts. Also number of Sperner systems with 2 blocks.

Original entry on oeis.org

1, 9, 55, 285, 1351, 6069, 26335, 111645, 465751, 1921029, 7859215, 31964205, 129442951, 522538389, 2104469695, 8460859965, 33972448951, 136276954149, 546269553775, 2188563950925, 8764714059751, 35090233104309, 140455067207455, 562102681589085, 2249257981411351
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Half the number of 2 X (n+2) binary arrays with both a path of adjacent 1's and a path of adjacent 0's from top row to bottom row. - R. H. Hardin, Mar 21 2002
As (0,0,1,9,55,...) this is the third binomial transform of cosh(x)-1. It is the binomial transform of A000392, when this has two leading zeros. Its e.g.f. is then exp(3x)cosh(x) - exp(3x) and a(n) = (4^n - 2*3^n + 2^n)/2. - Paul Barry, May 13 2003
Let P(A) be the power set of an n-element set A. Then a(n-2) is the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x. - Ross La Haye, Jan 10 2008
a(n) also gives the third column sequence of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below, and comments on the general case under A193685. - Wolfdieter Lang, Oct 08 2011
a(n) is also the number of even binomial coefficients in rows 0 through 2^(n+1)-1 of Pascal's triangle. - Aaron Meyerowitz, Oct 29 2013

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 292, #8, s(n,2).

Links

Crossrefs

Equals (1/2) A038721(n+1). First differences of A000453. Partial sums of A027650. Pairwise sums of A099110. Odd part of A019333.
Cf. A000079, A000392, A001047, A006516, A032263, A143494, A193685.

Programs

Formula

G.f.: 1/((1-2*x)*(1-3*x)*(1-4*x)).
a(n-2) = (2^n)*(2^n - 1)/2 - 3^n + 2^n.
From Hieronymus Fischer, Jun 25 2007: (Start)
a(n) = Sum_{0<=i,j,k,<=n, i+j+k=n} 2^i*3^j*4^k.
a(n) = 2^(n+1)*(1+2^(n+2))-3^(n+2). (End)
a(n) = 3*StirlingS2(n+3,4) + StirlingS2(n+3,3). - Ross La Haye, Jan 10 2008
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-2) = f(n,2,2), (n >= 2). - Milan Janjic, Apr 26 2009
E.g.f.: (d^2/dx^2) (exp(2*x)*((exp(x)-1)^2)/2!). See the Sheffer comment given above. - Wolfdieter Lang, Oct 08 2011
a(n) = A006516(n+2) - A001047(n+2). - Ross La Haye, Jan 26 2016
a(n) = A006516(n+1) + 3*a(n-1), n>=1, a(0)=1. - Carlos A. Rico A., Jun 22 2019

A017389 Expansion of g.f. 1/((1-3*x)*(1-5*x)*(1-7*x)).

Original entry on oeis.org

1, 15, 154, 1350, 10891, 83685, 623764, 4558380, 32875381, 234980955, 1669192174, 11806149810, 83252603071, 585817587825, 4115974729384, 28888095527640, 202598073849961, 1420093671872295, 9950191865139394, 69699025028403870, 488131588547752051, 3418113197039242365
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A019333.

Programs

  • Magma
    I:=[1,15,154]; [n le 3 select I[n] else 15*Self(n-1)-71*Self(n-2)+105*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jun 26 2013
  • Magma
    m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-5*x)*(1-7*x)))); // Vincenzo Librandi, Jun 26 2013
  • Maple
    A017389:=n->(7^(n+2) - 2*5^(n+2) + 3^(n+2))/8: seq(A017389(n), n=0..20); # Wesley Ivan Hurt, Mar 25 2014
  • Mathematica
    CoefficientList[Series[1 / ((1 - 3 x) (1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 26 2013 *)
  • PARI
    a(n) = (7^(n+2) - 2*5^(n+2) + 3^(n+2))/8; \\ Joerg Arndt, Aug 13 2013
  • PARI
    x='x+O('x^20); Vec(1/((1-3*x)*(1-5*x)*(1-7*x))) \\ Altug Alkan, Sep 23 2018

Formula

From Vincenzo Librandi, Jun 26 2013: (Start)
a(n) = 15*a(n-1) - 71*a(n-2) + 105*a(n-3).
a(n) = 12*a(n-1) - 35*a(n-2) + 3^n. (End)
a(n) = (7^(n+2) - 2*5^(n+2) + 3^(n+2))/8. - Yahia Kahloune, Aug 13 2013
From Seiichi Manyama, May 04 2025: (Start)
a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = Sum_{k=0..n} (-2)^k * 7^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)
E.g.f.: exp(3*x)*(9 - 50*exp(2*x) + 49*exp(4*x))/8. - Stefano Spezia, May 04 2025

A019943 Expansion of 1/((1-5*x)*(1-7*x)*(1-9*x)).

Original entry on oeis.org

1, 21, 298, 3570, 38971, 401751, 3988468, 38583300, 366449941, 3434404281, 31873887838, 293663563830, 2690806228111, 24553315831611, 223338364450408, 2026585451393160, 18355202849805481, 166009125098571741, 1499772036736668178, 13537796780062999290
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A017389, A019333.

Programs

  • GAP
    List([0..20], n -> (5^(n+2) - 2*7^(n+2) + 9^(n+2))/8); # G. C. Greubel, Nov 24 2018
  • Magma
    m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-5*x)*(1-7*x)*(1-9*x)))); // Vincenzo Librandi, Jul 03 2013
  • Magma
    I:=[1, 21, 298]; [n le 3 select I[n] else 21*Self(n-1)-143*Self(n-2)+315*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 03 2013
  • Magma
    [(5^(n+2) - 2*7^(n+2) + 9^(n+2))/8: n in [0..20]]; // G. C. Greubel, Nov 24 2018
  • Maple
    a:= n-> (Matrix(3, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [21, -143, 315][i], 0)))^n)[1, 1]: seq(a(n), n=0..25);  # Alois P. Heinz, Jul 03 2013
  • Mathematica
    CoefficientList[Series[1/((1-5x)(1-7x)(1-9 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 03 2013 *)
LinearRecurrence[{21,-143,315}, {1, 21, 298}, 20] (* or *) Table[(5^(n+2) - 2*7^(n+2) + 9^(n+2))/8, {n,0,20}] (* G. C. Greubel, Nov 24 2018 *)
  • PARI
    vector(20, n, n--; (5^(n+2) - 2*7^(n+2) + 9^(n+2))/8) \\ G. C. Greubel, Nov 24 2018
  • Sage
    [(5^(n+2) - 2*7^(n+2) + 9^(n+2))/8 for n in range(20)] # G. C. Greubel, Nov 24 2018

Formula

a(n) = (5^(n+2) - 2*7^(n+2) + 9^(n+2))/8. - R. J. Mathar, Jun 29 2013
a(0)=1, a(1)=21, a(2)=298; for n>2, a(n) = 21*a(n-1) -143*a(n-2) +315*a(n-3). - Vincenzo Librandi, Jul 03 2013
a(n) = 16*a(n-1) -63*a(n-2) +5^n. - Vincenzo Librandi, Jul 03 2013
E.g.f.: (25*exp(5*x) - 98*exp(7*x) + 81*exp(9*x))/8. - G. C. Greubel, Nov 24 2018
From Seiichi Manyama, May 04 2025: (Start)
a(n) = Sum_{k=0..n} 2^k * 5^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)

A021814 Expansion of 1/((1-x)(1-4x)(1-6x)(1-8x)).

Original entry on oeis.org

1, 19, 239, 2519, 24135, 218343, 1903783, 16194343, 135426599, 1118993447, 9166829607, 74629521447, 604827848743, 4885462331431, 39365093814311, 316610553147431, 2543028967600167, 20405121901817895
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A019333 (first differences).

Programs

  • Magma
    m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1-4*x)*(1-6*x)*(1-8*x)))); // Bruno Berselli, May 08 2013
  • Mathematica
    CoefficientList[Series[1/((1 - x) (1 - 4 x) (1 - 6 x) (1 - 8 x)), {x, 0, 20}], x] (* Bruno Berselli, May 08 2013 *)
  • PARI
    Vec(1/((1-x)*(1-4*x)*(1-6*x)*(1-8*x))+O(x^20)) \\ Bruno Berselli, May 08 2013

Formula

G.f.: 1/((1-x)*(1-4*x)*(1-6*x)*(1-8*x)).
a(n) = -1/105 +2^(2n+3)/3 -2^(n+1)*3^(n+3)/5 +8^(n+2)/7. [Bruno Berselli, May 08 2013]
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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