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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A005019 The number of n X n (0,1)-matrices with a 1-width of 1.

Original entry on oeis.org

1, 7, 169, 14911, 4925281, 6195974527, 30074093255809, 568640725896660991, 42170765737391337500161, 12325140160135610565932361727, 14244006984657003076298588475598849
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

a(n) is the number of ways to linearly order (with repetition allowed) n subsets of {1,2,...n} so that the generalized intersection of the subsets is not empty. - Geoffrey Critzer, Mar 01 2009
a(n) is the number of n X n binary matrices with at least one row of 0's. - Geoffrey Critzer, Dec 03 2009

Examples

			a(2)=7 because there are seven ways to order two subsets of {1,2} so that the intersection of the subsets contains at least one element: {1}{1};{1}{1,2};{2}{2};{2}{1,2};{1,2}{1};{1,2}{2};{1,2}{1,2}. - _Geoffrey Critzer_, Mar 01 2009

References

  • Lam, Clement W. H., The distribution of 1-widths of (0, 1)-matrices. Discrete Math. 20 (1977/78), no. 2, 109-122.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • Stanley, Enumerative Combinatorics, Volume I, Example 1.1.16 [From Geoffrey Critzer, Dec 03 2009]

Links

Crossrefs

a(n) = 2^(n^2)- A055601. - Geoffrey Critzer, Dec 03 2009
Cf. A005020 (1-width of 2).

Programs

  • Mathematica
    Table[2^(n^2) - (2^n - 1)^n, {n, 1, 15}] (* Geoffrey Critzer, Dec 03 2009 *)

Formula

a(n) = 2^(n^2) - ((2^n)-1)^n. - Geoffrey Critzer, Mar 01 2009

Extensions

a(7) from Geoffrey Critzer, Mar 01 2009
More terms from Geoffrey Critzer, Dec 03 2009
Title improved by Sean A. Irvine, Mar 06 2020

A202991 E.g.f: Sum_{n>=0} 3^(n^2) * exp(-2*3^n*x) * x^n/n!.

Original entry on oeis.org

1, 1, 49, 15625, 38950081, 812990017201, 147640825624179889, 237771659632917369765625, 3425319186561140076700951192321, 443021141828981570872668681812345111521, 515202988063835984513918825523304657054713360049
Offset: 0

Views

Author

Paul D. Hanna, Dec 26 2011

Keywords

Comments

E.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n/n! = Sum_{n>=0} (m*q^n + b)^n * x^n/n! for all q, m, b.
O.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * x^n/(1-b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n for all q, m, b.

Examples

			E.g.f.: A(x) = 1 + x + 49*x^2/2! + 15625*x^3/3! + 38950081*x^4/4! +...
By the series identity, the g.f.:
A(x) = exp(-2*x) + 3*exp(-2*3*x)*x + 3^4*exp(-2*3^2*x)*x^2/2! + 3^9*exp(-2*3^3*x)*x^3/3! + 3^16*exp(-2*3^4*x)*x^4/4! +...
expands into:
A(x) = 1 + x + 7^2*x^2/2! + 25^3*x^3/3! + 79^4*x^4/4! + 241^5*x^5/5! +...+ (3^n-2)^n*x^n/n! +...

Crossrefs

Cf. A180602, A165327, A202990, A202989, A060613, A055601.

Programs

  • PARI
    {a(n, q=3, m=1, b=-2)=(m*q^n + b)^n}
  • PARI
    {a(n, q=3, m=1, b=-2)=n!*polcoeff(sum(k=0, n, m^k*q^(k^2)*exp(b*q^k*x+x*O(x^n))*x^k/k!), n)}
  • PARI
    {a(n, q=3, m=1, b=-2)=polcoeff(sum(k=0, n, m^k*q^(k^2)*x^k/(1-b*q^k*x+x*O(x^n))^(k+1)), n)}

Formula

a(n) = (3^n - 2)^n.
O.g.f.: Sum_{n>=0} 3^(n^2)*x^n/(1 + 2*3^n*x)^(n+1).
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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