A342111 -id:A342111 - cp's OEIS frontend

A047796 a(n) = Sum_{k=0..n} Stirling1(n,k)^2.

1, 1, 2, 14, 194, 4402, 147552, 6838764, 418389078, 32639603798, 3161107700156, 372023906062756, 52280302234036252, 8645773770675973804, 1661888635268695003484, 367390786215560629372920, 92552610850186107484661670, 26356304249588730696338349990

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi and Vaclav Kotesovec, Table of n, a(n) for n = 0..250 (terms 0..41 from Vincenzo Librandi)

Cf. A000275, A047797, A342111.

List([0..20], n-> Sum([0..n], k-> Stirling1(n,k)^2 )); # G. C. Greubel, Aug 07 2019

[(&+[StirlingFirst(n,k)^2: k in [0..n]]): n in [0..10]]; // G. C. Greubel, Aug 07 2019

seq(add(stirling1(n, k)^2, k = 0..n), n = 0..20); # G. C. Greubel, Aug 07 2019

Table[Sum[StirlingS1[n,k]^2,{k,0,n}],{n,0,20}] (* Emanuele Munarini, Jul 04 2011 *)

makelist(sum(stirling1(n,k)^2,k,0,n),n,0,24); /* Emanuele Munarini, Jul 04 2011 */

a(n) = sum(k=0, n, stirling(n, k, 1)^2); \\ Michel Marcus, Mar 26 2016

[sum(stirling_number1(n,k)^2 for k in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 07 2019

A238261 Decimal expansion of a constant related to A187235.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Feb 21 2014

			4.9108149645682558987515348...

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.261.

Cf. A187235, A238262, A002465, A129256, A342111, A330287.

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

Equals lim n->infinity (A187235(n)/(n-1)!)^(1/n).

Equals -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)).

A384029 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^4.

Original entry on oeis.org

1, 0, 6, 180, 7206, 370880, 23477380, 1768061064, 154544373158, 15387101825184, 1719596420272980, 213181689525888600, 29036623040055512332, 4310582688852993653568, 692756995680614782818992, 119830419866883597939018000, 22198322332579642585088580870, 4384714751330840129324051474880

Offset: 0

Seiichi Manyama, May 17 2025

nonn

Cf. A342111, A384018.
Cf. A384027, A384030.
Cf. A384031.

a(n) = sum(i=0, n, sum(j=0, 3*n-i, sum(k=0, 3*n-i-j, abs(stirling(n, i, 1)*stirling(n, j, 1)*stirling(n, k, 1)*stirling(n, 3*n-i-j-k, 1)))));

a(n) = Sum_{0<=i, j, k, l<=n and i+j+k+l=3*n} |Stirling1(n,i) * Stirling1(n,j) * Stirling1(n,k) * Stirling1(n,l)|.

A342110 a(n) = Sum_{k=0..n} Stirling2(n,k) * Stirling2(n,n-k).

Original entry on oeis.org

1, 0, 1, 6, 61, 770, 12160, 228382, 4989621, 124262532, 3475892685, 107901412520, 3681266754660, 136918473752216, 5513911474915116, 239034083286873630, 11098790133822288645, 549539910028075555016, 28903562131933534643851, 1609321474965547356327246

Offset: 0

Vaclav Kotesovec, Feb 28 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A008277, A014322, A047797, A342111.

Cf. A048993.

[(&+[StirlingSecond(n, k)*StirlingSecond(n, n-k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 03 2021

Table[Sum[StirlingS2[n, k]*StirlingS2[n, n-k], {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, stirling(n, k, 2)*stirling(n, n-k, 2)); \\ Michel Marcus, Feb 28 2021

[sum( stirling_number2(n, k)*stirling_number2(n, n-k) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Jun 03 2021

From Vaclav Kotesovec, Feb 28 2021, updated May 25 2025: (Start)
a(n) ~ c * d^n * (n-1)!, where
d = A238258 = -2 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 3.0882773047417401791158400820254382768364448971420138767247...
c = 1/(2*Pi*sqrt((1 + LambertW(-2*exp(-2)))*(3 + LambertW(-2*exp(-2))))) = 0.12826577250734152801558828593238744179869387423941684693208180123477... (End)

A384018 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^3.

Original entry on oeis.org

1, 0, 3, 63, 1767, 63690, 2822740, 148810032, 9104502015, 634448680884, 49622704133175, 4305280182748875, 410376649359397380, 42633179822414174760, 4794685285831034253660, 580373328155358031572600, 75234419898396217903091151, 10398952352945773993329785448, 1526704288048697734221906020641

Offset: 0

Seiichi Manyama, May 17 2025

nonn

Cf. A342111, A384029.

Cf. A384026.

a(n) = sum(i=0, n, sum(j=0, 2*n-i, abs(stirling(n, i, 1)*stirling(n, j, 1)*stirling(n, 2*n-i-j, 1))));

a(n) = Sum_{0<=i, j, k<=n and i+j+k=2*n} |Stirling1(n,i) * Stirling1(n,j) * Stirling1(n,k)|.

A384027 a(n) = [x^(3n)] Product_{k=0..n-1} (1 + kx)^4.

Original entry on oeis.org

1, 0, 0, 0, 1296, 2764800, 8041766400, 34726710251520, 219045033712578816, 1956771788423009992704, 24009126017002632247173120, 393692515265172002272138690560, 8424620140673205407840209386541056, 230472036551670538296109810120063451136, 7917891968134805796965854747528387122954240

Offset: 0

Seiichi Manyama, May 17 2025

nonn

Cf. A342111, A384026.

Cf. A384029, A384030.

a(n) = sum(i=0, n, sum(j=0, n-i, sum(k=0, n-i-j, abs(stirling(n, i, 1)*stirling(n, j, 1)*stirling(n, k, 1)*stirling(n, n-i-j-k, 1)))));

a(n) = Sum_{i, j, k, l>=0 and i+j+k+l=n} |Stirling1(n,i) * Stirling1(n,j) * Stirling1(n,k) * Stirling1(n,l)|.

A155826 Triangle T(n, k) = binomial(n, k) + binomial(k(n-k), n) + 2(-1)^nStirlingS1(n, k)StirlingS1(n, n-k), read by rows.

Original entry on oeis.org

4, 1, 1, 1, 4, 1, 1, 15, 15, 1, 1, 76, 249, 76, 1, 1, 485, 3516, 3516, 485, 1, 1, 3606, 46623, 101354, 46623, 3606, 1, 1, 30247, 617541, 2388107, 2388107, 617541, 30247, 1, 1, 282248, 8416315, 51483931, 91651662, 51483931, 8416315, 282248, 1, 1, 2903049, 119667766, 1071669632, 3021085118, 3021085118, 1071669632, 119667766, 2903049, 1

Offset: 0

Roger L. Bagula, Jan 28 2009

			Triangle begins as:
  4;
  1,      1;
  1,      4,       1;
  1,     15,      15,        1;
  1,     76,     249,       76,        1;
  1,    485,    3516,     3516,      485,        1;
  1,   3606,   46623,   101354,    46623,     3606,       1;
  1,  30247,  617541,  2388107,  2388107,   617541,   30247,     1;
  1, 282248, 8416315, 51483931, 91651662, 51483931, 8416315, 282248, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A048994, A342111.

A155826:= func< n,k | Binomial(n, k) + Binomial(k*(n-k), n) + 2*(-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k) >;
[A155826(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 03 2021

T[n_, k_]:= Binomial[n, k] + Binomial[k*(n-k), n] + 2*(-1)^n*StirlingS1[n, k]*StirlingS1[n, n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 03 2021 *)

def A155826(n,k): return binomial(n, k) + binomial(k*(n-k), n) + 2*stirling_number1(n, k)*stirling_number1(n, n-k)
flatten([[A155826(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 03 2021

T(n, k) = binomial(n, k) + binomial(k*(n-k), n) + 2*(-1)^n*StirlingS1(n, k) * StirlingS1(n, n-k).

Sum_{k=0..n} T(n, k) = 2^n + 2*342111(n) + Sum_{k=0..n} binomial(k*(n-k), n). - G. C. Greubel, Jun 03 2021

Edited by G. C. Greubel, Jun 03 2021

A383880 a(n) = [x^n] 1/Product_{k=0..n-1} (1 - k*x)^2.

Original entry on oeis.org

1, 0, 3, 72, 2307, 95060, 4817990, 290523576, 20333487251, 1621036680120, 145057745669850, 14399349523416000, 1570425994090538574, 186674663305762642296, 24021930409036829669036, 3327140929951823209016400, 493515678917684006649451651, 78054583374364036172432641200

Offset: 0

Seiichi Manyama, May 13 2025

nonn

Cf. A350376, A383883.

Cf. A342111.

Join[{1}, Table[Sum[StirlingS2[n + k - 1, n - 1]*StirlingS2[2*n - k - 1, n - 1], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 14 2025 *)

a(n) = polcoef(1/prod(k=0, n-1, 1-k*x+x*O(x^n))^2, n);

a(n) = Sum_{k=0..n} Stirling2(n+k-1,n-1) * Stirling2(2*n-k-1,n-1) for n > 0.

a(n) ~ 3^(3*n - 3/2) * n^(n - 1/2) / (sqrt(Pi*(1-w)) * 2^(2*n - 1/2) * exp(n) * (3 - 2*w)^n * w^(2*n - 3/2)), where w = -LambertW(-3*exp(-3/2)/2). - Vaclav Kotesovec, May 14 2025

A384026 a(n) = [x^(2n)] Product_{k=0..n-1} (1 + kx)^3.

Original entry on oeis.org

1, 0, 0, 8, 1188, 240480, 68630824, 26730127872, 13715719388784, 8994742935058880, 7351374493516431744, 7333037983443263351040, 8772990646534399559904256, 12403600039078715891159873280, 20464777911173655904724421045504, 38976211807455406964301439206318080

Offset: 0

Seiichi Manyama, May 17 2025

nonn

Cf. A342111, A384027.

Cf. A384018.

a(n) = sum(i=0, n, sum(j=0, n-i, abs(stirling(n, i, 1)*stirling(n, j, 1)*stirling(n, n-i-j, 1))));

a(n) = Sum_{i, j, k>=0 and i+j+k=n} |Stirling1(n,i) * Stirling1(n,j) * Stirling1(n,k)|.

A155742 Triangle T(n, k) = (-1)^nStirlingS1(n, k)StirlingS1(n, n-k), read by rows.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 6, 6, 0, 0, 36, 121, 36, 0, 0, 240, 1750, 1750, 240, 0, 0, 1800, 23290, 50625, 23290, 1800, 0, 0, 15120, 308700, 1193640, 1193640, 308700, 15120, 0, 0, 141120, 4207896, 25738720, 45819361, 25738720, 4207896, 141120, 0, 0, 1451520, 59832864, 535810464, 1510458516, 1510458516, 535810464, 59832864, 1451520, 0

Offset: 0

Roger L. Bagula, Jan 26 2009

			Triangle begins as:
  1;
  0,      0;
  0,      1,       0;
  0,      6,       6,        0;
  0,     36,     121,       36,        0;
  0,    240,    1750,     1750,      240,        0;
  0,   1800,   23290,    50625,    23290,     1800,       0;
  0,  15120,  308700,  1193640,  1193640,   308700,   15120,      0;
  0, 141120, 4207896, 25738720, 45819361, 25738720, 4207896, 141120, 0;

G. C. Greubel, Rows n = 0..50 of the triangle flattened

Cf. A048994, A342111 (row sums).

A155742:= func< n,k | (-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k) >;
[A155742(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 05 2021

T[n_, k_]:= (-1)^n*StirlingS1[n, k]*StirlingS1[n, n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 05 2021 *)

def A155742(n,k): return stirling_number1(n,k)*stirling_number1(n, n-k)
flatten([[A155742(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 05 2021

T(n, k) = (-1)^n*StirlingS1(n, k)*StirlingS1(n, n-k).

Sum_{k=0..n} T(n, k) = A342111(n). - G. C. Greubel, Jun 05 2021

Edited by G. C. Greubel, Jun 05 2021

cp's OEIS Frontend

A047796 a(n) = Sum_{k=0..n} Stirling1(n,k)^2.

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A238261 Decimal expansion of a constant related to A187235.

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A384029 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^4.

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A342110 a(n) = Sum_{k=0..n} Stirling2(n,k) * Stirling2(n,n-k).

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A384018 a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^3.

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A384027 a(n) = [x^(3*n)] Product_{k=0..n-1} (1 + k*x)^4.

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A155826 Triangle T(n, k) = binomial(n, k) + binomial(k*(n-k), n) + 2*(-1)^n*StirlingS1(n, k)*StirlingS1(n, n-k), read by rows.

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A383880 a(n) = [x^n] 1/Product_{k=0..n-1} (1 - k*x)^2.

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A384027 a(n) = [x^(3n)] Product_{k=0..n-1} (1 + kx)^4.

A155826 Triangle T(n, k) = binomial(n, k) + binomial(k(n-k), n) + 2(-1)^nStirlingS1(n, k)StirlingS1(n, n-k), read by rows.

A384026 a(n) = [x^(2n)] Product_{k=0..n-1} (1 + kx)^3.

A155742 Triangle T(n, k) = (-1)^nStirlingS1(n, k)StirlingS1(n, n-k), read by rows.