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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).

%I A226513 #55 Feb 03 2024 10:13:56
%S A226513 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,
%T A226513 2612,4683,1,7,48,305,1704,7803,25988,47293,1,8,63,468,3155,18424,
%U A226513 87135,296564,545835,1,9,80,679,5340,37625,227304,1102419,3816548,7087261
%N A226513 Array read by antidiagonals: T(n,k) = number of barred preferential arrangements of k things with n bars (k >=0, n >= 0).
%C A226513 The terms of this sequence are also called high-order Fubini numbers (see p. 255 in Komatsu). - _Stefano Spezia_, Dec 06 2020
%D A226513 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.
%H A226513 Alois P. Heinz, <a href="/A226513/b226513.txt">Antidiagonals n = 0..140, flattened</a>
%H A226513 Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, <a href="https://doi.org/10.37236/2482">Barred Preferential Arrangements</a>, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.
%H A226513 Toka Diagana and Hamadoun Maïga, <a href="https://doi.org/10.1016/j.jnt.2016.09.032">Some new identities and congruences for Fubini numbers</a>, J. Number Theory 173 (2017), 547-569.
%H A226513 Takao Komatsu, <a href="https://www.researchgate.net/publication/344595540_SHIFTED_BERNOULLI_NUMBERS_AND_SHIFTED_FUBINI_NUMBERS">Shifted Bernoulli numbers and shifted Fubini numbers</a>, Linear and Nonlinear Analysis, Volume 6, Number 2, 2020, 245-263 (p. 255).
%F A226513 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
%F A226513 T(n,k) = k! * [x^k] 1/(2-exp(x))^(n+1). - _Alois P. Heinz_, Mar 26 2016
%F A226513 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
%F A226513 From _Seiichi Manyama_, Nov 19 2023: (Start)
%F A226513 T(n,0) = 1; T(n,k) = Sum_{j=1..k} (n*j/k + 1) * binomial(k,j) * T(n,k-j).
%F A226513 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)
%F A226513 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
%e A226513 Array begins:
%e A226513   1  1   3   13    75    541     4683     47293     545835 ...
%e A226513   1  2   8   44   308   2612    25988    296564    3816548 ...
%e A226513   1  3  15   99   807   7803    87135   1102419   15575127 ...
%e A226513   1  4  24  184  1704  18424   227304   3147064   48278184 ...
%e A226513   1  5  35  305  3155  37625   507035   7608305  125687555 ...
%e A226513   1  6  48  468  5340  69516  1014348  16372908  289366860 ...
%e A226513   ...
%e A226513 Triangle begins:
%e A226513   1,
%e A226513   1, 1,
%e A226513   1, 2, 3,
%e A226513   1, 3, 8, 13,
%e A226513   1, 4, 15, 44, 75,
%e A226513   1, 5, 24, 99, 308, 541,
%e A226513   1, 6, 35, 184, 807, 2612, 4683,
%e A226513   1, 7, 48, 305, 1704, 7803, 25988, 47293,
%e A226513   1, 8, 63, 468, 3155, 18424, 87135, 296564, 545835
%e A226513   ........
%e A226513 [_Vincenzo Librandi_, Jun 18 2013]
%p A226513 T:= (n, k)-> k!*coeff(series(1/(2-exp(x))^(n+1), x, k+1), x, k):
%p A226513 seq(seq(T(d-k, k), k=0..d), d=0..10);  # _Alois P. Heinz_, Mar 26 2016
%t A226513 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 *)
%Y A226513 Rows 0-5 give: A000670, A005649, A226515, A226738, A226739, A226740.
%Y A226513 Columns 2, 3 = A005563, A226514.
%Y A226513 Cf. A053492 (array diagonal), A265609, A346982.
%K A226513 nonn,tabl
%O A226513 0,5
%A A226513 _N. J. A. Sloane_, Jun 13 2013