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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A189843 Number of ways to place n nonattacking composite pieces rook + semi-rider[2,2] on an n X n chessboard.

Original entry on oeis.org

1, 2, 5, 18, 71, 356, 2097, 14212, 105821, 887576, 8093601, 81310936, 876456695, 10257217440, 127631146697, 1705775408656
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

a(n) is also number of permutations p of 1,2,...,n satisfying p(j+2k)-p(j)<>2k for all j>=1, k>=1, j+2k<=n
about semi-pieces see semi-bishop (A187235) and semi-queen (A099152)

Links

Crossrefs

Cf. A189281, A099152, A189838, A187235

A238258 Decimal expansion of a constant related to A002465.

Original entry on oeis.org

3, 0, 8, 8, 2, 7, 7, 3, 0, 4, 7, 4, 1, 7, 4, 0, 1, 7, 9, 1, 1, 5, 8, 4, 0, 0, 8, 2, 0, 2, 5, 4, 3, 8, 2, 7, 6, 8, 3, 6, 4, 4, 4, 8, 9, 7, 1, 4, 2, 0, 1, 3, 8, 7, 6, 7, 2, 4, 7, 7, 3, 0, 1, 2, 1, 7, 6, 5, 1, 6, 8, 1, 2, 7, 8, 8, 2, 6, 6, 6, 6, 9, 5, 2, 0, 3, 2, 7, 1, 1, 3, 0, 9, 6, 1, 9, 4, 6, 0, 0, 9, 7, 3, 0, 9
Offset: 1

Views

Author

Vaclav Kotesovec, Feb 21 2014

Keywords

Examples

			3.08827730474174017911584...

Links

Crossrefs

Cf. A002465, A238260, A187235, A226775, A342110.

Programs

  • Mathematica
    RealDigits[N[-2/LambertW[-2/E^2]/(2+LambertW[-2/E^2]), 105]][[1]]

Formula

Equals lim n->infinity (A002465(n)/(n-1)!)^(1/n).
Equals -2 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))).
Equals -2 / (A226775 * (2 + A226775)).

A189844 Number of ways to place n nonattacking composite pieces rook + semi-rider[3,3] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 22, 98, 534, 3334, 23724, 191820, 1704532, 16689868, 179288892, 2069311996, 25760882744, 345073745880, 4900331447624
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

a(n) is also number of permutations p of 1,2,...,n satisfying p(j+3k)-p(j)<>3k for all j>=1, k>=1, j+3k<=n.
For information about semi-pieces see semi-bishop (A187235) and semi-queen (A099152).

Links

Crossrefs

Cf. A189282, A099152, A189843, A189839, A187235

A254881 Triangle read by rows, T(n,k) = sum(j=0..k-1, S(n+1,j+1)*S(n,k-j)) where S denotes the Stirling cycle numbers A132393, T(0,0)=1, n>=0, 0<=k<=2n.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 5, 4, 1, 0, 12, 40, 51, 31, 9, 1, 0, 144, 564, 904, 769, 376, 106, 16, 1, 0, 2880, 12576, 23300, 24080, 15345, 6273, 1650, 270, 25, 1, 0, 86400, 408960, 840216, 991276, 748530, 381065, 133848, 32523, 5370, 575, 36, 1, 0, 3628800, 18299520
Offset: 0

Views

Author

Peter Luschny, Feb 10 2015

Keywords

Comments

These are also the coefficients of the polynomials interpolating the sequence k -> n!*((n+k)!/k!)*binomial(n+k-1,k-1) (for fixed n>=0). Divided by n! these polynomials generate the rows of Lah numbers L(n+k, k) = ((n+k)!/k!)* binomial(n+k-1,k-1).

Examples

			[1]
[0, 1, 1]
[0, 2, 5, 4, 1]
[0, 12, 40, 51, 31, 9, 1]
[0, 144, 564, 904, 769, 376, 106, 16, 1]
[0, 2880, 12576, 23300, 24080, 15345, 6273, 1650, 270, 25, 1]
For example in the case n=3 the polynomial (k^6+9*k^5+31*k^4+51*k^3+40*k^2+12*k)/3! generates the Lah numbers 0, 24, 240, 1200, 4200, 11760, 28224, ... (A253285).

Crossrefs

Cf. A246117, A254882, A187235.
The sequences A000012, A002378, A083374, A253285 are the Lah number rows generated by the polynomials divided by n! for n=0, 1, 2, 3 respectivly.

Programs

  • Maple
    # This is a special case of the recurrence given in A246117.
t := proc(n,k) option remember; if n=0 and k=0 then 1 elif
k <= 0 or k>n then 0 else iquo(n,2)*t(n-1,k)+t(n-1,k-1) fi end:
A254881 := (n,k) -> t(2*n,k):
seq(print(seq(A254881(n,k), k=0..2*n)), n=0..5);
# Illustrating the comment:
restart: with(PolynomialTools): with(CurveFitting): for N from 0 to 5 do
CoefficientList(PolynomialInterpolation([seq([k,N!*((N+k)!/k!)*binomial(N+k-1,k-1)], k=0..2*N)], n), n) od;
  • Mathematica
    Flatten[{1,Table[Table[Sum[Abs[StirlingS1[n+1,j+1]] * Abs[StirlingS1[n,k-j]],{j,0,k-1}],{k,0,2*n}],{n,1,10}]}] (* Vaclav Kotesovec, Feb 10 2015 *)
  • Sage
    def T(n,k):
    if n == 0: return 1
    return sum(stirling_number1(n+1,j+1)*stirling_number1(n,k-j) for j in range(k))
for n in range (6): [T(n,k) for k in (0..2*n)]

Formula

T(n, n) = A187235(n) for n>=1 (after the explicit formula of Vaclav Kotesovec).

A182562 Number of ways to place k non-attacking semi-knights on an n x n chessboard, sum over all k>=0.

Original entry on oeis.org

2, 16, 288, 11664, 1458000, 506250000, 414720000000, 869730877440000, 5045702916833280000, 77297454895962562560000, 3017525202366485003182080000, 307389127582207654481154908160000, 83016370640108703579427655610531840000, 58770343311359208383258439665073059266560000
Offset: 1

Views

Author

Vaclav Kotesovec, May 05 2012

Keywords

Comments

Semi-knight is a semi-leaper [1,2]. Moves of a semi-knight are allowed only in [2,1] and [-2,-1]. See also semi-bishops (A187235).

Links

Crossrefs

Cf. A067962, A067966, A063443, A006506, A067965, A066864, A067963, A067964, A182563

Programs

  • Mathematica
    Table[If[EvenQ[n],Fibonacci[n/2+2]^(n+2)*Product[Fibonacci[j+2]^4,{j,1,n/2-1}],Fibonacci[(n+1)/2+2]^((n+1)/2)*Fibonacci[(n-1)/2+2]^((n-1)/2)*Product[Fibonacci[j+2]^4,{j,1,(n-1)/2}]],{n,1,20}]

Formula

a(n) = F(n/2+2)^(n+2)*prod(j=1,n/2-1,F(j+2)^4) if n is even, F((n+1)/2+2)^((n+1)/2)*F((n-1)/2+2)^((n-1)/2)*prod(j=1,(n-1)/2,F(j+2)^4) if n is odd, where F(n) = A000045(n) is the n-th Fibonacci number.
a(n) is asymptotic to C^4*((1+sqrt(5))/2)^((n+2)*(n+4))/5^(3/2*(n+2)), where C=1.226742010720353244... is Fibonacci Factorial Constant, see A062073.

A189846 Number of ways to place n nonattacking composite pieces rook + semi-rider[4,4] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 114, 628, 4062, 30360, 251658, 2308648, 23351268, 259031232, 3091784268, 39697601392, 546982720164, 8064677125440
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

a(n) is also number of permutations p of 1,2,...,n satisfying p(j+4k)-p(j)<>4k for all j>=1, k>=1, j+4k<=n
For information about semi-pieces see semi-bishop (A187235) and semi-queen (A099152).

Links

Crossrefs

Cf. A189283, A099152, A189843, A189844, A189840, A187235

A238260 Decimal expansion of a multiplicative constant related to A002465.

Original entry on oeis.org

6, 3, 1, 2, 6, 6, 8, 7, 8, 8, 7, 4, 1, 1, 5, 4, 6, 7, 9, 7, 0, 0, 4, 8, 2, 3, 2, 5, 7, 9, 7, 0, 6, 8, 7, 9, 5, 5, 6, 1, 5, 4, 6, 9, 0, 5, 1, 4, 4, 6, 1, 1, 4, 0, 8, 9, 2, 0, 0, 6, 9, 7, 3, 4, 0, 5, 0, 8, 5, 4, 1, 5, 0, 3, 7, 6, 6, 1, 7, 0, 8, 5, 6, 0, 4, 0, 0, 8, 5, 0, 1, 7, 6, 1, 1, 0, 9, 3, 3, 5, 4, 6, 3, 5, 5
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 21 2014

Keywords

Examples

			0.63126687887411546797...

Links

Crossrefs

Cf. A002465, A238258, A187235.

Formula

Equals lim n->infinity A002465(n) / ((n-1)! * A238258^n).

A383883 a(n) = [x^n] 1/((1 - n*x) * Product_{k=0..n-1} (1 - k*x)^2).

Original entry on oeis.org

1, 1, 11, 222, 6627, 262570, 12978758, 769079444, 53138842515, 4194648739710, 372421403333850, 36733739199892020, 3985122473105099406, 471598870326072262644, 60456151456891375730860, 8345905345383943433713800, 1234395864446065862689721475, 194738649118647202909304657910
Offset: 0

Views

Author

Seiichi Manyama, May 13 2025

Keywords

Crossrefs

Cf. A350376, A383880.
Cf. A187235, A287532.

Programs

  • Mathematica
    Join[{1}, Table[Sum[StirlingS2[n + k - 1, n - 1]*StirlingS2[2*n - k, n], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 14 2025 *)
  • PARI
    a(n) = polcoef(1/((1-n*x)*prod(k=0, n-1, 1-k*x+x*O(x^n))^2), n);

Formula

a(n) = Sum_{k=0..n} Stirling2(n+k-1,n-1) * Stirling2(2*n-k,n) for n > 0.
a(n) = A287532(n,n).
a(n) ~ 3^(3*n - 1/2) * n^(n - 1/2) / (sqrt(Pi*(1-w)) * 2^(2*n + 1/2) * exp(n) * (3 - 2*w)^n * w^(2*n - 1/2)), where w = -LambertW(-3*exp(-3/2)/2). - Vaclav Kotesovec, May 14 2025

A182563 Number of ways to place n non-attacking semi-knights on an n x n chessboard.

Original entry on oeis.org

1, 6, 70, 1289, 33864, 1148760, 47700972, 2344465830, 133055587660, 8559364525414, 615266768106190, 48861588247978827, 4247584874013608724, 401107335066453376830, 40880928693752664368224, 4472281486633326131737868
Offset: 1

Views

Author

Vaclav Kotesovec, May 05 2012

Keywords

Comments

Semi-knight is a semi-leaper [1,2]. Moves of a semi-knight are allowed only in [2,1] and [-2,-1]. See also semi-bishops (A187235).

Links

Crossrefs

Cf. A182562, A201540, A201513, A002465, A201511, A197989, A201861.

Formula

Asymptotic: a(n) ~ n^(2n)/n!*exp(-3/2).

Extensions

a(16) from Vaclav Kotesovec, May 24 2021

A189847 Number of ways to place n nonattacking composite pieces rook + semi-rider[5,5] on an n X n chessboard.

Original entry on oeis.org

1, 2, 6, 24, 120, 696, 4572, 34260, 290328, 2751480, 28426056, 318900264, 3874868280, 50813711808, 716309557440, 10721493269568
Offset: 1

Views

Author

Vaclav Kotesovec, Apr 29 2011

Keywords

Comments

a(n) is also number of permutations p of 1,2,...,n satisfying p(j+5k)-p(j)<>5k for all j>=1, k>=1, j+5k<=n
For information about semi-pieces see semi-bishop (A187235) and semi-queen (A099152).

Links

Crossrefs

Cf. A189284, A099152, A189843, A189844, A189846, A189841, A187235
Previous Showing 11-20 of 22 results. Next

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