A226514 -id:A226514 - 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

A253286 Square array read by upward antidiagonals, A(n,k) = Sum_{j=0..n} (n-j)!C(n,n-j) C(n-1,n-j)*k^j, for n>=0 and k>=0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 13, 8, 3, 1, 0, 73, 44, 15, 4, 1, 0, 501, 304, 99, 24, 5, 1, 0, 4051, 2512, 801, 184, 35, 6, 1, 0, 37633, 24064, 7623, 1696, 305, 48, 7, 1, 0, 394353, 261536, 83079, 18144, 3145, 468, 63, 8, 1

Offset: 0

Peter Luschny, Mar 24 2015

			Square array starts, A(n,k):
      1,       1,       1,       1,      1,      1,      1, ...  A000012
      0,       1,       2,       3,      4,      5,      6, ...  A001477
      0,       3,       8,      15,     24,     35,     48, ...  A005563
      0,      13,      44,      99,    184,    305,    468, ...  A226514
      0,      73,     304,     801,   1696,   3145,   5328, ...
      0,     501,    2512,    7623,  18144,  37225,  68976, ...
      0,    4051,   24064,   83079, 220096, 495475, 997056, ...
A000007, A000262, A052897, A255806, ...
Triangle starts, T(n, k) = A(n-k, k):
  1;
  0,   1;
  0,   1,   1;
  0,   3,   2,  1;
  0,  13,   8,  3,  1;
  0,  73,  44, 15,  4, 1;
  0, 501, 304, 99, 24, 5, 1;

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Laguerre polynomials

Main diagonal gives A293145.

Cf. A000262, A001477, A005563, A052897, A226514, A255806, A256325.

[k eq n select 1 else k*Factorial(n-k-1)*Evaluate(LaguerrePolynomial(n-k-1, 1), -k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021

L := (n, k) -> (n-k)!*binomial(n,n-k)*binomial(n-1,n-k):
A := (n, k) -> add(L(n,j)*k^j, j=0..n):
# Alternatively:
# A := (n, k) -> `if`(n=0,1, simplify(k*n!*hypergeom([1-n],[2],-k))):
for n from 0 to 6 do lprint(seq(A(n,k), k=0..6)) od;

A253286[n_, k_]:= If[k==n, 1, k*(n-k-1)!*LaguerreL[n-k-1, 1, -k]];
Table[A253286[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)

{T(n, k) = if(n==0, 1, n!*sum(j=1, n, k^j*binomial(n-1, j-1)/j!))} \\ Seiichi Manyama, Feb 03 2021

{T(n, k) = if(n<2, (k-1)*n+1, (2*n+k-2)*T(n-1, k)-(n-1)*(n-2)*T(n-2, k))} \\ Seiichi Manyama, Feb 03 2021

flatten([[1 if k==n else k*factorial(n-k-1)*gen_laguerre(n-k-1, 1, -k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 23 2021

A(n,k) = k*n!*hypergeom([1-n],[2],-k) for n>=1 and 1 for n=0.
Row sums of triangle, Sum_{k=0..n} A(n-k, k) = 1 + A256325(n).
From Seiichi Manyama, Feb 03 2021: (Start)
E.g.f. of column k: exp(k*x/(1-x)).
T(n,k) = (2*n+k-2) * T(n-1,k) - (n-1) * (n-2) * T(n-2, k) for n > 1. (End)
From G. C. Greubel, Feb 23 2021: (Start)
A(n, k) = k*(n-1)!*LaguerreL(n-1, 1, -k) with A(0, k) = 1.
T(n, k) = k*(n-k-1)!*LaguerreL(n-k-1, 1, -k) with T(n, n) = 1.
T(n, 2) = A052897(n) = A086915(n)/2.
Sum_{k=0..n} T(n, k) = 1 + Sum_{k=0..n-1} (n-k-1)*k!*LaguerreL(k, 1, k-n+1). (End)

A226741 Column 4 of array in A226513.

Original entry on oeis.org

75, 308, 807, 1704, 3155, 5340, 8463, 12752, 18459, 25860, 35255, 46968, 61347, 78764, 99615, 124320, 153323, 187092, 226119, 270920, 322035, 380028, 445487, 519024, 601275, 692900, 794583, 907032, 1030979, 1167180, 1316415, 1479488, 1657227, 1850484

Offset: 0

Vincenzo Librandi, Jun 18 2013

This is the case h = 4 in Sum_{k=0..h} S2(h,k)*k!*binomial(n+k,k), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3). [Bruno Berselli, Jun 20 2013]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. columns 2, 3 and 5, 6 of A226513: A005563, A226514, A226800, A226801.

[(n+1)^4+12*(n+1)^3+36*(n+1)^2+26*(n+1): n in [0..35]];

I:=[75, 308, 807, 1704, 3155]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]];

Table[(n+1)^4 + 12 (n+1)^3 + 36 (n+1)^2 + 26 (n+1), {n, 0, 40}] (* or *) CoefficientList[Series[(75 - 67 x + 17 x^2 - x^3) / (1 - x)^5, {x, 0, 40}], x]

G.f.: (75 - 67*x + 17*x^2 - x^3)/(1 - x)^5.
a(n) = (n + 1)^4 + 12*(n + 1)^3 + 36*(n + 1)^2 + 26*(n + 1).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
E.g.f.: exp(x)*(75 + 233*x + 133*x^2 + 22*x^3 + x^4). - Franck Maminirina Ramaharo, Nov 29 2018

A226800 Column 5 of array in A226513.

Original entry on oeis.org

541, 2612, 7803, 18424, 37625, 69516, 119287, 193328, 299349, 446500, 645491, 908712, 1250353, 1686524, 2235375, 2917216, 3754637, 4772628, 5998699, 7463000, 9198441, 11240812, 13628903, 16404624, 19613125, 23302916, 27525987, 32337928, 37798049, 43969500

Offset: 0

Vincenzo Librandi, Jun 18 2013

This is the case h = 5 in Sum_{k=0..h} S2(h,k)*k!*binomial(n+k,k), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3). [Bruno Berselli, Jun 20 2013]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. columns 2, 3, 4 and 6 of A226513: A005563, A226514, A226741, A226801.

[(n+1)^5+20*(n+1)^4+120*(n+1)^3+250*(n+1)^2+150*(n+1): n in [0..30]];

I:=[541,2612,7803,18424,37625,69516]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..30]];

Table[(n + 1)^5 + 20 (n+1)^4 + 120 (n+1)^3 + 250 (n+1)^2 + 150 (n+1), {n, 0, 40}] (* or *) CoefficientList[Series[(541 - 634 x + 246 x^2 - 34 x^3 + x^4) / (1 - x)^6, {x, 0, 30}], x]

G.f.: (541 - 634*x + 246*x^2 - 34*x^3 + x^4)/(1 - x)^6.
a(n) = (n + 1)*(n^4 + 24*n^3 + 186*n^2 + 554*n + 541).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
E.g.f.: exp(x)*(541 + 2071*x + 1560*x^2 + 385*x^3 + 35*x^4 + x^5). - Franck Maminirina Ramaharo, Nov 29 2018

A226801 Column 6 of array in A226513.

Original entry on oeis.org

4683, 25988, 87135, 227304, 507035, 1014348, 1871583, 3242960, 5342859, 8444820, 12891263, 19103928, 27595035, 38979164, 53985855, 73472928, 98440523, 130045860, 169618719, 218677640, 278946843, 352373868, 441147935, 547719024, 674817675, 825475508

Offset: 0

Vincenzo Librandi, Jun 20 2013

This is the case h = 6 in Sum_{k=0..h} S2(h,k)*k!*binomial(n+k,k), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3). [Bruno Berselli, Jun 20 2013]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. columns 2, 3, 4, 5 of A226513: A005563, A226514, A226741, A226800.

[(n+1)*(n^5+35*n^4+430*n^3+2320*n^2+5525*n+4683): n in [0..35]];

I:=[4683,25988,87135,227304,507035,1014348,1871583]; [n le 7 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..30]];

Table[(n + 1) (n^5 + 35 n^4 + 430 n^3 + 2320 n^2 + 5525 n + 4683), {n, 0, 40}] (* or *) CoefficientList[Series[(4683 - 6793 x + 3562 x^2 - 798 x^3 + 67 x^4 - x^5) / (1-x)^7, {x, 0, 30}], x]
LinearRecurrence[{7,-21,35,-35,21,-7,1},{4683,25988,87135,227304,507035,1014348,1871583},30] (* Harvey P. Dale, Apr 27 2014 *)

G.f.: (4683 - 6793*x + 3562*x^2 - 798*x^3 + 67*x^4 - x^5) / (1-x)^7.
a(n) = (n + 1)*(n^5 + 35*n^4 + 430*n^3 + 2320*n^2 + 5525*n + 4683).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
E.g.f.: exp(x)*(4683 + 21305*x + 19921*x^2 + 6530*x^3 + 890*x^4 + 51*x^5 + x^6). - Franck Maminirina Ramaharo, Nov 29 2018

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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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A253286 Square array read by upward antidiagonals, A(n,k) = Sum_{j=0..n} (n-j)!*C(n,n-j)* C(n-1,n-j)*k^j, for n>=0 and k>=0.

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A226741 Column 4 of array in A226513.

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A226800 Column 5 of array in A226513.

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A226801 Column 6 of array in A226513.

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A253286 Square array read by upward antidiagonals, A(n,k) = Sum_{j=0..n} (n-j)!C(n,n-j) C(n-1,n-j)*k^j, for n>=0 and k>=0.