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 1-3 of 3 results.

A099259 A100287(n+1) - A000960(n).

Original entry on oeis.org

1, 2, 2, 2, 6, 4, 4, 12, 4, 8, 12, 14, 6, 22, 2, 12, 18, 6, 12, 24, 12, 4, 48, 4, 30, 24, 22, 30, 38, 14, 20, 12, 24, 4, 30, 18, 6, 68, 38, 48, 26, 34, 8, 56, 24, 6, 40, 56, 10, 86, 6, 2, 76, 42, 72, 48, 26, 4, 40, 52, 12, 54, 12, 8, 46, 26, 24, 72, 12, 42, 74, 40, 36, 98, 22, 18, 58
Offset: 1

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Author

N. J. A. Sloane, Nov 16 2004

Keywords

Comments

Always positive.

Crossrefs

Cf. A000960, A100002, A099264.

Extensions

More terms from David Wasserman, Feb 27 2008

A099265 Partial sums of A056272.

Original entry on oeis.org

1, 3, 8, 23, 75, 277, 1132, 4977, 22979, 109451, 531456, 2610931, 12917683, 64181625, 319695980, 1594859885, 7963472187, 39784944799, 198827606704, 993846943839, 4968361974491, 24839192686973, 124188113975628, 620917025694793, 3104514504312595, 15522360665856147, 77611167795714752
Offset: 1

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Author

Nelma Moreira, Oct 10 2004

Keywords

Comments

Density of regular language L{0}* over {0, 1, 2, 3, 4, 5} (i.e., the number of strings of length n), where L is described by regular expression with c = 5: Sum_{i=1..c} Product_{j=1..i} (j(1+...+j)*), where "Sum" stands for union and "Product" for concatenation. I.e., L = L((11* + 11*2(1 + 2)* + ... + 11*2(1 + 2)*3(1 + 2 + 3)*4(1 + 2 + 3 + 4)*5(1 + 2 + 3 + 4 + 5)*)0*).

Links

Crossrefs

Cf. A008277, A047926, A056272, A099264, A099266.

Programs

  • Maple
    with (combinat):seq(sum(sum(stirling2(k, j),j=1..5), k=1..n), n=1..23); # Zerinvary Lajos, Dec 04 2007
  • PARI
    a(n) = sum(m=1, n, sum(i=1, 5, stirling(m, i, 2))) \\ Petros Hadjicostas, Mar 10 2021

Formula

a(5,n) = (1/96)*5^n + (1/8)*3^n + (1/3)*2^n + (3/8)*n - 15/32.
a(n) = Sum_{m=1..n} Sum_{i=1..5} S(m,i), where S(m,i) = A008277(m,i) (i.e., partial sum of the sum of Stirling numbers of second kind S(n,i) for i = 1..5).
For c = 5, a(c,n) = g(1,c)*n + Sum_{k=2..c} g(k,c)*k*(k^n - 1)/(k - 1), where g(1,1) = 1, g(1,c) = g(1,c-1) + (-1)^(c-1)/(c-1)! for c > 1, and g(k,c) = g(k-1, c-1)/k for c > 1 and 2 <= k <= c.
G.f.: x*(-1 + 19*x^3 - 24*x^2 + 9*x)/((3*x-1)*(2*x-1)*(5*x-1)*(x-1)^2). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009]

Extensions

Name and Formula section edited by Petros Hadjicostas, Mar 10 2021
More terms from Michel Marcus, Jan 05 2025

A099266 Partial sums of A056273.

Original entry on oeis.org

1, 3, 8, 23, 75, 278, 1154, 5265, 25913, 135212, 736704, 4139831, 23767895, 138468210, 814675838, 4824766301, 28699128501, 171207852152, 1023332115836, 6124430348355, 36684624841811, 219860794899518, 1318179574171578
Offset: 1

Views

Author

Nelma Moreira, Oct 10 2004

Keywords

Comments

Some previous names were a(6,n) := (1/600)*6^n + (1/36)*4^n + (1/12)*3^n + (3/8)*2^n + (11/30)*n - (439/900) = Sum_{m=1..n} Sum_{i=1..6} S(m,i), where S(n,i) = A008277(n,i) are the Stirling numbers of the second kind.
Density of the regular language L{0}* over {0, 1, 2, 3, 4, 5, 6} (i.e., the number of strings of length n), where L is described by regular expression with c = 6: Sum_{i=1..c} Prod_{j=1..i} (j(1+...+j)*), where "Sum" stands for union and "Product" for concatenation. I.e., L = L((11* + ... + 11*2(1 + 2)*3(1 + 2 + 3)*4(1 + 2 + 3 + 4)*5(1 + 2 + 3 + 4 + 5)*6(1 + 2 + 3 + 4 + 5 + 6)*)0*).

Links

Crossrefs

Cf. A047926, A056273, A008277, A099264, A099265.

Programs

  • Maple
    with (combinat):seq(sum(sum(stirling2(k, j),j=1..6), k=1..n), n=1..23); # Zerinvary Lajos, Dec 04 2007
  • PARI
    Vec(x*(91*x^4-135*x^3+68*x^2-14*x+1)/((x-1)^2*(2*x-1)*(3*x-1)*(4*x-1)*(6*x-1)) + O(x^100)) \\ Colin Barker, Oct 28 2014
  • PARI
    a(n) = sum(m=1, n, sum(i=1, 6, stirling(m, i, 2))) \\ Petros Hadjicostas, Mar 09 2021

Formula

For c = 6, a(c, n) = g(1, c)*n + Sum_{k=2..c} g(k, c)*k*(k^n - 1)/(k - 1), where g(1, 1) = 1, g(1, c) = g(1, c-1) + (-1)^(c-1)/(c-1)! for c > 1, and g(k, c) = g(k-1, c-1)/k for c > 1 and 2 <= k <= c.
G.f.: x*(91*x^4 - 135*x^3 + 68*x^2 - 14*x + 1) / ((x - 1)^2*(2*x - 1)*(3*x - 1)*(4*x - 1)*(6*x - 1)). - Colin Barker, Oct 28 2014

Extensions

Shorter name by Joerg Arndt, Oct 28 2014
Comments edited by Petros Hadjicostas, Mar 09 2021
Showing 1-3 of 3 results.

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

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