A187535 -id:A187535 - cp's OEIS frontend

A187666 Binomial convolution of the (signless) central Lah numbers (A187535) and the (signless) central Stirling numbers of the first kind (A187646).

1, 3, 51, 1599, 74545, 4654255, 365549495, 34642467783, 3846064986001, 489429448820811, 70208261310969435, 11205444535728231855, 1969021774778391995761, 377672618542009829524551, 78507169034687468202172591

Offset: 0

Emanuele Munarini, Mar 12 2011

Cf. A187535, A187646.

L := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(sum(binomial(n,k)*L(k)*abs(combinat[stirling1](2*(n-k),n-k)),k=0..n),n=0..12);

L[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[Binomial[n,k]L[k]Abs[StirlingS1[2n - 2k, n - k]], {k, 0, n}], {n, 0, 14}]

L(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(binomial(n,k)*L(k)*abs(stirling1(2*n-2*k,n-k)),k,0,n),n,0,12);

a(n) = Sum_{k=0..n} binomial(n,k) * Lah(2k,k) * Stirling1(2n-2k,n-k).

a(n) ~ c * 2^(4*n + 1/2) * n^(n - 1/2) / (sqrt(Pi) * exp(n)), where c = Sum_{k>=0} abs(Stirling1(2*k,k)) / (k! * 2^(4*k+1)) = 0.550990257867992515027936630097897... - Vaclav Kotesovec, May 30 2025

A187548 Alternating partial sums of L(n)*H(n+1), product of central Lah number L(n) and Harmonic number H(n+1).

Original entry on oeis.org

1, 2, 64, 2436, 131824, 9203264, 787735648, 79884060128, 9366719620672, 1246887723480128, 185786630586649792, 30635253866287585088, 5538860010787064796352, 1089574788981508858403648, 231683608824013918904796352, 52954849085008593516185123648

Offset: 0

Emanuele Munarini, Mar 11 2011

Cf. A187535, A001008, A187547.

H := proc(n) add(1/i,i=1..n) ; end proc:
A187535 := proc(n) if n=0 then 1; else binomial(2*n-1, n-1)*(2*n)!/n! end if; end proc:
A187547 := proc(n) H(n+1)*A187535(n) ; end proc:
A187548 := proc(n) add( A187547(k)*(-1)^(n-k),k=0..n) ; end proc:
seq(A187548(n),n=0..20) ; # R. J. Mathar, Mar 24 2011

Table[Sum[(-1)^n+(-1)^(n-k)Binomial[2k-1,k-1](2k)!/k!HarmonicNumber[k+1],{k,1,n}],{n,0,12}]

makelist((-1)^n+sum((-1)^(n-k)*binomial(2*k-1,k-1)*(2*k)!/k!*sum(1/i,i,1,k+1),k,1,n),n,0,12);

a(n) = Sum_{k=0..n} (-1)^(n-k)*A187547(k).

A248045 (2(n-1))! (2n-1)! / (n (n-1)!^3).

Original entry on oeis.org

1, 6, 120, 4200, 211680, 13970880, 1141620480, 111307996800, 12614906304000, 1629845894476800, 236475822507724800, 38072607423743692800, 6735922851893114880000, 1299070835722243584000000, 271245990498804460339200000, 60962536364606302461235200000

Offset: 1

Reinhard Zumkeller, Sep 30 2014

nonn

Central terms in triangles of Lah numbers: a(n) = - A008297(2*n-1,n) = A105278(2*n-1,n) = A000891(n-1)*A000142(n) = A000894(n-1)*A000142(n-1).

a(n) = n * A204515(n-1). - Reinhard Zumkeller, Oct 19 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A000142, A000891, A000894, A008297, A105278, A204515.

Cf. A187535 (Central Lah numbers).

Haskell
```
a248045 n = a000891 (n - 1) * a000142 n
```

n*a(n) = 4*(2*n-1)*(2*n-3)*a(n-1). - R. J. Mathar, Oct 07 2014

A123072 Bishops on an 8n+1 X 8n+1 board (see Robinson paper for details).

Original entry on oeis.org

1, 2, 72, 7200, 1411200, 457228800, 221298739200, 149597947699200, 134638152929280000, 155641704786247680000, 224746621711341649920000, 396453040698806670458880000, 838894634118674914690990080000, 2097236585296687286727475200000000, 6115541882725140128097317683200000000

Offset: 0

N. J. A. Sloane, Sep 28 2006

nonn

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence zeta(2k+1).]

Cf. A173331. [Reinhard Zumkeller, Feb 16 2010]

Cf. A001700, A010050, A156992, A187535.

Maple
```
For Maple program see A005635.
```

Table[(((2 n)!/n!)^2)/2, {n, 1, 20}] (* Benedict W. J. Irwin, Jun 05 2016 *)
Table[SeriesCoefficient[Series[1/2 + EllipticK[16 x]/Pi, {x, 0, 20}],n] n! n!, {n, 1, 20}] (* Benedict W. J. Irwin, Jun 05 2016 *)

From_Reinhard Zumkeller_, Feb 16 2010: (Start)
a(n) = ceiling((((2*n)! / n!)^2) / 2).
a(n) = A001700(n-1) * A010050(n). (End)
From Benedict W. J. Irwin, Jun 05 2016: (Start)
G.f. for a(n)/(n!)^2 : 1/2 + EllipticK(16*x)/Pi, which is the E.g.f for A187535.
G.f. for a(n)/(n!)^3 : 2F2(1/2, 1/2; 1, 1; 16z)/2.
a(n) = n!*A187535(n) = binomial(2*n-1, n-1)*(2*n)!.
(End)
a(n) = A156992(2n,n). - Alois P. Heinz, Apr 30 2017
a(n) ~ asy(2*n-1) where asy(n) = (2*n/e)^n*(18*n + 6 + 1/n)/9. - Peter Luschny, Dec 05 2019
Sum_{n>=0} 1/a(n) = 1 + StruveL(0, 1/2)*Pi/4, where StruveL is the modified Struve function. - Amiram Eldar, Dec 04 2022

a(0)=1 prepended by Alois P. Heinz, Apr 30 2017

A187665 Binomial convolution of the central Lah numbers and the central Stirling numbers of the second kind.

Original entry on oeis.org

1, 3, 47, 1440, 67533, 4280175, 341307292, 32750424588, 3670267277749, 470237282353989, 67781221867781615, 10855095004543985756, 1912103925425230231884, 367398970712627913234708, 76469792506315229551855080

Offset: 0

Emanuele Munarini, Mar 12 2011

A048993 := proc(n,k) combinat[stirling2](n, k) ; end proc:
A187535 := proc(n) if n=0 then 1 else binomial(2*n-1, n-1)*(2*n)!/n! end if; end proc:
A187665 := proc(n) add(binomial(n,k)*A187535(k)*A048993(2*n-2*k,n-k), k=0..n) ; end proc:
seq(A187665(n),n=0..10)  ; # R. J. Mathar, Mar 28 2011

L[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[Binomial[n,k]L[k]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 14}]

L(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(binomial(n,k)*L(k)*stirling2(2*n-2*k,n-k),k,0,n),n,0,12);

a(n) = Sum_{k=0..n} binomial(n,k)*A187535(k)* A048993(2n-2k,n-k).

a(n) ~ c * 16^n * (n-1)!, where c = (1/Pi) * Sum_{k>=0} abs(Stirling2(2*k,k)) / (k! * 2^(4*k+1)) = 0.172113078600558193773... - Vaclav Kotesovec, Jul 05 2021, updated May 30 2025

A355004 a(n) = Sum_{k=0..n} A271703(k + n, n), row sums of A355005.

Original entry on oeis.org

1, 3, 43, 1333, 63321, 4034341, 321994723, 30869387193, 3454384526353, 441903886812721, 63608031487665171, 10174227287873082853, 1790258521269694523113, 343669522619597368671933, 71473405251333054552561091, 16008271911444915765782477041, 3841639137772270982094393928353

Offset: 0

Peter Luschny, Jun 15 2022

nonn

Cf. A271703 (unsigned Lah), A355005, A187535.

L := (n, k) -> ifelse(n = k, 1, binomial(n-1, k-1)*n! / k!):
seq(add(L(n + k, n), k = 0..n), n = 0..16);

Table[Sum[Binomial[n + k, n]*FactorialPower[n + k - 1, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 15 2022 *)

a(n) = A187535(n) * hypergeom([1, -n], [1 - 2*n, -2*n], -1).
From Vaclav Kotesovec, Jun 15 2022: (Start)
Recurrence: (n-1)^2 * n * (64*n^4 - 464*n^3 + 1244*n^2 - 1475*n + 663)*a(n) = (n-1)*(2*n-3)*(512*n^6 - 3968*n^5 + 11872*n^4 - 17336*n^3 + 12880*n^2 - 4597*n + 617)*a(n-1) + (2048*n^7 - 19968*n^6 + 78912*n^5 - 163216*n^4 + 191140*n^3 - 128857*n^2 + 48842*n - 8937)*a(n-2) + 4*(2*n-5)*(2*n-3)*(64*n^4 - 208*n^3 + 236*n^2 - 123*n + 32)*a(n-3).
a(n) ~ 2^(4*n - 1/2) * n^(n - 1/2) / (sqrt(Pi) * exp(n)). (End)

A355005 Table read by rows. T(n, k) = n((k + n)!)^2/((k + n)(n!)^2*k!) for n > 0 and T(0, 0) = 1.

Original entry on oeis.org

1, 1, 2, 1, 6, 36, 1, 12, 120, 1200, 1, 20, 300, 4200, 58800, 1, 30, 630, 11760, 211680, 3810240, 1, 42, 1176, 28224, 635040, 13970880, 307359360, 1, 56, 2016, 60480, 1663200, 43908480, 1141620480, 29682132480, 1, 72, 3240, 118800, 3920400, 122316480, 3710266560, 111307996800, 3339239904000

Offset: 0

Peter Luschny, Jun 15 2022

			[0] 1;
[1] 1,  2;
[2] 1,  6,   36;
[3] 1, 12,  120,  1200;
[4] 1, 20,  300,  4200,   58800;
[5] 1, 30,  630, 11760,  211680,  3810240;
[6] 1, 42, 1176, 28224,  635040, 13970880,  307359360;
[7] 1, 56, 2016, 60480, 1663200, 43908480, 1141620480, 29682132480;

T(n, 1) = A002378, T(n, n) = A187535, A355004 (row sums), A271703 (Lah).

T := (n, k) -> ifelse(n = 0, 1, n*((k + n)!)^2 / ((k + n)*(n!)^2*k!)):
seq(seq(T(n, k), k = 0..n), n = 0..8);

T(n, k) = Lah(k + n, n), where Lah denotes the unsigned Lah numbers A271703.

A361893 Triangle read by rows. T(n, k) = n! * binomial(n - 1, k - 1) / (n - k)!.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 12, 6, 0, 4, 36, 72, 24, 0, 5, 80, 360, 480, 120, 0, 6, 150, 1200, 3600, 3600, 720, 0, 7, 252, 3150, 16800, 37800, 30240, 5040, 0, 8, 392, 7056, 58800, 235200, 423360, 282240, 40320, 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880

Offset: 0

Peter Luschny, Mar 28 2023

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 2,   2;
  [3] 0, 3,  12,     6;
  [4] 0, 4,  36,    72,     24;
  [5] 0, 5,  80,   360,    480,     120;
  [6] 0, 6, 150,  1200,   3600,    3600,     720;
  [7] 0, 7, 252,  3150,  16800,   37800,   30240,    5040;
  [8] 0, 8, 392,  7056,  58800,  235200,  423360,  282240,   40320;
  [9] 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880;

Cf. A052852 (row sums), A317365 (alternating row sums), A000142 (main diagonal), A187535 (central column), A062119, A055303, A011379.

Cf. A144084, A021010.

A361893 := (n, k) -> n!*binomial(n - 1, k - 1)/(n - k)!:
seq(seq(A361893(n,k), k = 0..n), n = 0..9);
# Using the egf.:
egf := 1 + (x*y/(1 - x*y))*exp(y/(1 - x*y)): ser := series(egf, y, 10):
poly := n -> convert(n!*expand(coeff(ser, y, n)), polynom):
row := n -> seq(coeff(poly(n), x, k), k = 0..n): seq(print(row(n)), n = 0..6);

T(n, k) = k! * binomial(n, k) * binomial(n - 1, k - 1).
T(n + 1, k + 1) / (n + 1) = A144084(n, k) = (-1)^(n - k)*A021010(n, k).
T(n, k) = [x^k] n! * ([y^n](1 + (x*y / (1 - x*y)) * exp(y / (1 - x*y)))).

A343581 a(n) = binomial(n, floor(n/2))*FallingFactorial(n - 1, n - floor(n/2)).

Original entry on oeis.org

1, 0, 2, 6, 36, 240, 1200, 12600, 58800, 846720, 3810240, 69854400, 307359360, 6849722880, 29682132480, 779155977600, 3339239904000, 100919250432000, 428906814336000, 14668613050291200, 61934143990118400, 2364758225077248000, 9931984545324441600, 418798681661180620800

Offset: 0

Peter Luschny, Apr 21 2021

nonn

Partially ordered sets on n elements that consist entirely of floor(n/2) chains (nonempty, linearly ordered subsets).

Cf. A187535, A271703.

a := n -> `if`(n=0, 1, binomial(n - 1, iquo(n,2) - 1)*n!/iquo(n, 2)!):
seq(a(n), n = 0..21);

a(n) = sum(j=n\2, n, abs(stirling(n, j, 1))*stirling(j, n\2, 2)); \\ Michel Marcus, Apr 22 2021

def a(n): return binomial(n, n - n//2)*falling_factorial(n - 1, n - n//2)
print([a(n) for n in range(22)])

a(n) = Sum_{j=floor(n/2)..n} |Stirling1(n, j)|*Stirling2(j, floor(n/2)).
a(n) = binomial(n - 1, floor(n/2) - 1)*n!/floor(n/2)! for n >= 1, a(0) = 1.
a(n) = A271703(n, floor(n/2)).

cp's OEIS Frontend

A187666 Binomial convolution of the (signless) central Lah numbers (A187535) and the (signless) central Stirling numbers of the first kind (A187646).

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A187548 Alternating partial sums of L(n)*H(n+1), product of central Lah number L(n) and Harmonic number H(n+1).

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A248045 (2*(n-1))! * (2*n-1)! / (n * (n-1)!^3).

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Haskell

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A123072 Bishops on an 8n+1 X 8n+1 board (see Robinson paper for details).

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Extensions

A187665 Binomial convolution of the central Lah numbers and the central Stirling numbers of the second kind.

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Maple

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Maxima

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A355004 a(n) = Sum_{k=0..n} A271703(k + n, n), row sums of A355005.

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A355005 Table read by rows. T(n, k) = n*((k + n)!)^2/((k + n)*(n!)^2*k!) for n > 0 and T(0, 0) = 1.

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A361893 Triangle read by rows. T(n, k) = n! * binomial(n - 1, k - 1) / (n - k)!.

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A343581 a(n) = binomial(n, floor(n/2))*FallingFactorial(n - 1, n - floor(n/2)).

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A248045 (2(n-1))! (2n-1)! / (n (n-1)!^3).

A355005 Table read by rows. T(n, k) = n((k + n)!)^2/((k + n)(n!)^2*k!) for n > 0 and T(0, 0) = 1.