A226515 -id:A226515 - cp's OEIS frontend

A226513 Array read by antidiagonals: T(n,k) = number of barred preferential arrangements of k things with n bars (k >=0, n >= 0).

1, 1, 1, 1, 2, 3, 1, 3, 8, 13, 1, 4, 15, 44, 75, 1, 5, 24, 99, 308, 541, 1, 6, 35, 184, 807, 2612, 4683, 1, 7, 48, 305, 1704, 7803, 25988, 47293, 1, 8, 63, 468, 3155, 18424, 87135, 296564, 545835, 1, 9, 80, 679, 5340, 37625, 227304, 1102419, 3816548, 7087261

Offset: 0

N. J. A. Sloane, Jun 13 2013

The terms of this sequence are also called high-order Fubini numbers (see p. 255 in Komatsu). - Stefano Spezia, Dec 06 2020

			Array begins:
  1  1   3   13    75    541     4683     47293     545835 ...
  1  2   8   44   308   2612    25988    296564    3816548 ...
  1  3  15   99   807   7803    87135   1102419   15575127 ...
  1  4  24  184  1704  18424   227304   3147064   48278184 ...
  1  5  35  305  3155  37625   507035   7608305  125687555 ...
  1  6  48  468  5340  69516  1014348  16372908  289366860 ...
  ...
Triangle begins:
  1,
  1, 1,
  1, 2, 3,
  1, 3, 8, 13,
  1, 4, 15, 44, 75,
  1, 5, 24, 99, 308, 541,
  1, 6, 35, 184, 807, 2612, 4683,
  1, 7, 48, 305, 1704, 7803, 25988, 47293,
  1, 8, 63, 468, 3155, 18424, 87135, 296564, 545835
  ........
[_Vincenzo Librandi_, Jun 18 2013]

Z.-R. Li, Computational formulae for generalized mth order Bell numbers and generalized mth order ordered Bell numbers (in Chinese), J. Shandong Univ. Nat. Sci. 42 (2007), 59-63.

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.
Toka Diagana and Hamadoun Maïga, Some new identities and congruences for Fubini numbers, J. Number Theory 173 (2017), 547-569.
Takao Komatsu, Shifted Bernoulli numbers and shifted Fubini numbers, Linear and Nonlinear Analysis, Volume 6, Number 2, 2020, 245-263 (p. 255).

Rows 0-5 give: A000670, A005649, A226515, A226738, A226739, A226740.
Columns 2, 3 = A005563, A226514.
Cf. A053492 (array diagonal), A265609, A346982.

T:= (n, k)-> k!*coeff(series(1/(2-exp(x))^(n+1), x, k+1), x, k):
seq(seq(T(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, Mar 26 2016

T[n_, k_] := Sum[StirlingS2[k, i]*i!*Binomial[n+i, i], {i, 0, k}]; Table[ T[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 26 2016 *)

T(n,k) = Sum_{i=0..k} S2_k(i)*i!*binomial(n+i,i), where S2_k(i) is the Stirling number of the second kind. - Jean-François Alcover, Mar 26 2016
T(n,k) = k! * [x^k] 1/(2-exp(x))^(n+1). - Alois P. Heinz, Mar 26 2016
Conjectural g.f. for row n as a continued fraction of Stieltjes type: 1/(1 - (n+1)*x/(1 - 2*x/(1 - (n+2)*x/(1 - 4*x/(1 - (n+3)*x/(1 - 6*x/(1 - ... ))))))). Cf. A265609. - Peter Bala, Aug 27 2023
From Seiichi Manyama, Nov 19 2023: (Start)
T(n,0) = 1; T(n,k) = Sum_{j=1..k} (n*j/k + 1) * binomial(k,j) * T(n,k-j).
T(n,0) = 1; T(n,k) = (n+1)*T(n,k-1) - 2*Sum_{j=1..k-1} (-1)^j * binomial(k-1,j) * T(n,k-j). (End)
G.f. for row n: (1/n!) * Sum_{m>=0} (n+m)! * x^m / Product_{j=1..m} (1 - j*x), for n >= 0. - Paul D. Hanna, Feb 01 2024

A354122 Expansion of e.g.f. 1/(1 + log(1 - x))^3.

Original entry on oeis.org

1, 3, 15, 102, 870, 8892, 105708, 1431168, 21722136, 365105928, 6729341832, 134915992560, 2922576142320, 68013701197920, 1692075061072800, 44810389419079680, 1258472984174461440, 37357062009383877120, 1168635883239630120960, 38424619272539153157120

Offset: 0

Seiichi Manyama, May 17 2022

nonn

Cf. A007840, A052801, A354123.

Cf. A226515, A354120.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x))^3))

a(n) = sum(k=0, n, (k+2)!*abs(stirling(n, k, 1)))/2;

a(n) = (1/2) * Sum_{k=0..n} (k + 2)! * |Stirling1(n,k)|.
a(n) ~ sqrt(Pi/2) * n^(n + 5/2) / (exp(1) - 1)^(n+3). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (2*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A226738 Row 3 of array in A226513.

Original entry on oeis.org

1, 4, 24, 184, 1704, 18424, 227304, 3147064, 48278184, 812387704, 14872295784, 294192418744, 6251984167464, 142032703137784, 3434617880825064, 88075274293319224, 2387099326339205544, 68177508876215724664, 2046501717592969431144, 64408432189100396344504

Offset: 0

Vincenzo Librandi, Jun 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.

Cf. rows 0, 1, 2, 4, 5 of A226513: A000670, A005649, A226515, A226739, A226740.

m:=3; [&+[StirlingSecond(n,i)*Factorial(i)*Binomial(m+i,i): i in [0..n]]: n in [0..20]]; // Bruno Berselli, Jun 18 2013

Range[0, 20]! CoefficientList[Series[(2 - Exp@x)^-4, {x, 0, 20}], x]

E.g.f.: 1/(2 - exp(x))^4 (see the Ahlbach et al. paper, Theorem 4).
a(n) = sum( S2(n,i)*i!*binomial(3+i,i), i=0..n ), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3). [Bruno Berselli, Jun 18 2013]
G.f.: 1/T(0), where T(k) = 1 - x*(k+4)/(1 - 2*x*(k+1)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 28 2013
a(n) ~ n! * n^3 / (96 * log(2)^(n+4)). - Vaclav Kotesovec, Oct 11 2022
Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 4*x/(1 - 6*x/(1 - 6*x/(1 - (n+3)*x/(1 - 2*n*x/(1 - ... ))))))))). - Peter Bala, Aug 27 2023
From Seiichi Manyama, Nov 19 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} (3*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)

A367471 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^3.

Original entry on oeis.org

1, 6, 54, 630, 8982, 150966, 2918934, 63772470, 1552910742, 41690570166, 1223096629014, 38924237638710, 1335418262833302, 49129420920630966, 1929262811804022294, 80540656071983191350, 3561781875173605408662, 166331104582900651581366

Offset: 0

Seiichi Manyama, Nov 19 2023

nonn

Cf. A004123, A367470.

Cf. A226515, A367473.

a(n) = sum(k=0, n, 2^k*(k+2)!*stirling(n, k, 2))/2;

a(n) = (1/2) * Sum_{k=0..n} 2^k * (k+2)! * Stirling2(n,k).
a(0) = 1; a(n) = 2*Sum_{k=1..n} (2*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 6*a(n-1) - 3*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k).

A226739 Row 4 of array in A226513.

Original entry on oeis.org

1, 5, 35, 305, 3155, 37625, 507035, 7608305, 125687555, 2265230825, 44210200235, 928594230305, 20880079975955, 500343586672025, 12726718227077435, 342425052939060305, 9715738272696568355, 289901469137229041225, 9074304882434034258635, 297297854264669632338305

Offset: 0

Vincenzo Librandi, Jun 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.

Cf. rows 0, 1, 2, 3 and 5 of A226513: A000670, A005649, A226515, A226738, A226740.

m:=4; [&+[StirlingSecond(n, i)*Factorial(i)*Binomial(m+i, i): i in [0..n]]: n in [0..20]];

Range[0,20]! CoefficientList[Series[(2 - Exp@x)^-5, {x, 0, 20}],x]

E.g.f.: 1/(2 - exp(x))^5 (see the Ahlbach et al. paper, Theorem 4).
a(n) = Sum_{i=0..n} S2(n,i)*i!*binomial(4+i,i), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3).
G.f.: 1/Q(0), where Q(k) = 1 - x*(3*k + 1 + m) - 2*x^2*(k + 1)*(k + 1 + m)/Q(k+1), m = 4 is row 4 of array in A226513; (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n) ~ n! * n^4 / (768 * log(2)^(n+5)). - Vaclav Kotesovec, Oct 11 2022
Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 5*x/(1 - 2*x/(1 - 6*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - (n+4)*x/(1 - 2*n*x/(1 - ... ))))))))). - Peter Bala, Aug 27 2023
From Seiichi Manyama, Nov 19 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} (4*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 5*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)

A226740 Row 5 of array in A226513.

Original entry on oeis.org

1, 6, 48, 468, 5340, 69516, 1014348, 16372908, 289366860, 5553635436, 114964523148, 2552305112748, 60474398655180, 1522843616043756, 40605864407444748, 1142786353739186988, 33848016050071188300, 1052381222812017946476, 34266937867683980363148, 1166071764343727862515628

Offset: 0

Vincenzo Librandi, Jun 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.

Cf. rows 0, 1, 2, 3, 4 of A226513: A000670, A005649, A226515, A226738, A226739.

m:=5; [&+[StirlingSecond(n, i)*Factorial(i)*Binomial(m+i, i): i in [0..n]]: n in [0..20]];

Range[0, 20]! CoefficientList[Series[(2 - Exp@x)^-6, {x, 0, 20}], x]

E.g.f.: 1/(2 - exp(x))^6 (see the Ahlbach et al. paper, Theorem 4).
a(n) = Sum_{i=0..n} S2(n,i)*i!*binomial(5+i,i), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3).
a(n) ~ n! * n^5 / (7680 * log(2)^(n+6)). - Vaclav Kotesovec, Oct 11 2022
Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 6*x/(1 - 2*x/(1 - 7*x/(1 - 4*x/(1 - 8*x/(1 - 6*x/(1 - (n+5)*x/(1 - 2*n*x/(1 - ... ))))))))). - Peter Bala, Aug 27 2023
From Seiichi Manyama, Nov 19 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} (5*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 6*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)

A354120 Expansion of e.g.f. 1/(1 - log(1 + x))^3.

Original entry on oeis.org

1, 3, 9, 30, 114, 492, 2388, 12912, 77016, 503112, 3570552, 27399600, 225729360, 1991996640, 18690559200, 186620451840, 1963991600640, 21914748541440, 255336518292480, 3155705206364160, 40209018105116160, 547746803311864320, 7525926332189130240

Offset: 0

Seiichi Manyama, May 17 2022

sign

a(34) is negative. - Vaclav Kotesovec, Jun 04 2022

Vaclav Kotesovec, Table of n, a(n) for n = 0..449

Cf. A006252, A317280, A354121.

Cf. A226515, A354122.

Table[Sum[(k+2)! * StirlingS1[n,k], {k,0,n}]/2, {n,0,35}] (* Vaclav Kotesovec, Jun 04 2022 *)
With[{nn=30},CoefficientList[Series[1/(1-Log[1+x])^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 16 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x))^3))

a(n) = sum(k=0, n, (k+2)!*stirling(n, k, 1))/2;

a(n) = (1/2) * Sum_{k=0..n} (k + 2)! * Stirling1(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (2 * k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A305919 a(n) = n! * [x^n] 1/(2 - exp(x))^n.

Original entry on oeis.org

1, 1, 8, 99, 1704, 37625, 1014348, 32300359, 1186399952, 49376357109, 2296400723220, 118031059900523, 6643848377509368, 406471060412884753, 26856124898028246044, 1905791887135240982415, 144563460111417997403040, 11673024609379676114380877, 999663240630210837032231460

Offset: 0

Ilya Gutkovskiy, Jun 14 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..350
N. J. A. Sloane, Transforms

Cf. A000670, A005649, A053492, A226513, A226515.

Table[n! SeriesCoefficient[1/(2 - Exp[x])^n, {x, 0, n}], {n, 0, 18}]
Table[SeriesCoefficient[Sum[Binomial[n + k - 1, k] k! x^k/Product[1 - j x, {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 18}]
Table[Sum[StirlingS2[n, k] Binomial[n + k - 1, k] k!, {k, 0, n}], {n, 0, 18}]

a(n) = [x^n] Sum_{k>=0} binomial(n+k-1,k)*k!*x^k/Product_{j=1..k} (1 - j*x).
a(n) = Sum_{k=0..n} Stirling2(n,k)*binomial(n+k-1,k)*k!.
a(n) ~ n! * c * ((1 + r)*(1 + 2*r))^n / sqrt(n), where r = (-1 + 1/(-1 + LambertW(2*exp(1))))/2 = 0.833964643008471735434624869020826957702396269585... is the root of the equation (2 + 1/r) * (1 + r*LambertW(-exp(-1/r)/r)) = 1 and c = 1/sqrt(2*Pi*(1 + LambertW(2*exp(1)))) = 0.258877607195571655640738032164006... Equivalently, a(n) ~ LambertW(2*exp(1))^n * n^n / (sqrt(1 + LambertW(2*exp(1))) * 2^n * exp(n) * (LambertW(2*exp(1)) - 1)^(2*n)). - Vaclav Kotesovec, Dec 15 2019, updated Mar 17 2024

A367473 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^3.

Original entry on oeis.org

1, 9, 117, 1953, 39645, 946089, 25926597, 801869553, 27618402285, 1048096422009, 43444114011477, 1952712851250753, 94592798546953725, 4912513525545837129, 272265236648295312357, 16039329591716508497553, 1000809252891040145821965

Offset: 0

Seiichi Manyama, Nov 19 2023

nonn

Cf. A032033, A367472.

Cf. A226515, A367471.

a(n) = sum(k=0, n, 3^k*(k+2)!*stirling(n, k, 2))/2;

a(n) = (1/2) * Sum_{k=0..n} 3^k * (k+2)! * Stirling2(n,k).
a(0) = 1; a(n) = 3*Sum_{k=1..n} (2*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 9*a(n-1) - 4*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k).

A375661 Expansion of e.g.f. 1 / (1 - x * (exp(x) - 1))^3.

Original entry on oeis.org

1, 0, 6, 9, 156, 735, 9738, 83181, 1129656, 13662459, 207281190, 3151269033, 54457383060, 980680471095, 19240001086530, 397345461622245, 8763618490102128, 203472380293912563, 4991552271140255838, 128517790560854181537, 3472936316648987980620

Offset: 0

Seiichi Manyama, Aug 23 2024

nonn

Cf. A052848, A375660.

Cf. A226515.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*(exp(x)-1))^3))

a(n) = n!*sum(k=0, n\2, (k+2)!*stirling(n-k, k, 2)/(n-k)!)/2;

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A052848.

a(n) = (n!/2) * Sum_{k=0..floor(n/2)} (k+2)! * Stirling2(n-k,k)/(n-k)!.

cp's OEIS Frontend

A226513 Array read by antidiagonals: T(n,k) = number of barred preferential arrangements of k things with n bars (k >=0, n >= 0).

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A354122 Expansion of e.g.f. 1/(1 + log(1 - x))^3.

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A226738 Row 3 of array in A226513.

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A367471 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^3.

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A226739 Row 4 of array in A226513.

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A226740 Row 5 of array in A226513.

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A354120 Expansion of e.g.f. 1/(1 - log(1 + x))^3.

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A305919 a(n) = n! * [x^n] 1/(2 - exp(x))^n.

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