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A108087 Array, read by antidiagonals, where A(n,k) = exp(-1)*Sum_{i>=0} (i+k)^n/i!.

%I A108087 #61 Sep 02 2024 19:54:00
%S A108087 1,1,1,2,2,1,5,5,3,1,15,15,10,4,1,52,52,37,17,5,1,203,203,151,77,26,6,
%T A108087 1,877,877,674,372,141,37,7,1,4140,4140,3263,1915,799,235,50,8,1,
%U A108087 21147,21147,17007,10481,4736,1540,365,65,9,1,115975,115975,94828,60814,29371,10427,2727,537,82,10,1
%N A108087 Array, read by antidiagonals, where A(n,k) = exp(-1)*Sum_{i>=0} (i+k)^n/i!.
%C A108087 The column for k=0 is A000110 (Bell or exponential numbers). The column for k=1 is A000110 starting at offset 1. The column for k=2 is A005493 (Sum_{k=0..n} k*Stirling2(n,k).). The column for k=3 is A005494 (E.g.f.: exp(3*z+exp(z)-1).). The column for k=4 is A045379 (E.g.f.: exp(4*z+exp(z)-1).). The row for n=0 is 1's sequence, the row for n=1 is the natural numbers. The row for n=2 is A002522 (n^2 + 1.). The row for n=3 is A005491 (n^3 + 3n + 1.). The row for n=4 is A005492.
%C A108087 Number of ways of placing n labeled balls into n+k boxes, where k of the boxes are labeled and the rest are indistinguishable. - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006
%C A108087 The column for k = -1 (not shown) is A000296 (Number of partitions of an n-set into blocks of size >1. Also number of cyclically spaced (or feasible) partitions.). - _Gerald McGarvey_, Oct 08 2006
%C A108087 Equals antidiagonals of an array in which (n+1)-th column is the binomial transform of n-th column, with leftmost column = the Bell sequence, A000110. - _Gary W. Adamson_, Apr 16 2009
%C A108087 Number of partitions of [n+k] where at least k blocks contain their own index element.  A(2,2) = 10: 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. - _Alois P. Heinz_, Jan 07 2022
%D A108087 F. Ruskey, Combinatorial Generation, preprint, 2001.
%H A108087 Alois P. Heinz, <a href="/A108087/b108087.txt">Antidiagonals n = 0..140, flattened</a>
%H A108087 I. Mezo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Mezo/mezo9.html">The r-Bell numbers</a>, J. Int. Seq. 14 (2011) # 11.1.1, Figure 1.
%H A108087 J. Riordan, <a href="/A001861/a001861_1.pdf">Letter, Oct 31 1977</a>, The array is on the second page.
%H A108087 F. Ruskey, <a href="https://page.math.tu-berlin.de/~felsner/SemWS17-18/Ruskey-Comb-Gen.pdf">Combinatorial Generation</a>, 2003.
%H A108087 F. Ruskey, <a href="http://www.cs.usyd.edu.au/~algo4301/Lexicographic.ps">Lexicographic Algorithms</a> [Broken link]
%F A108087 For n> 1, A(n, k) = k^n + sum_{i=0..n-2} A086659(n, i)*k^i. (A086659 is set partitions of n containing k-1 blocks of length 1, with e.g.f: exp(x*y)*(exp(exp(x)-1-x)-1).)
%F A108087 A(n, k) = k * A(n-1, k) + A(n-1, k+1), A(0, k) = 1. - Bradley Austin (artax(AT)cruzio.com), Apr 24 2006
%F A108087 A(n,k) = Sum_{i=0..n} C(n,i) * k^i * Bell(n-i). - _Alois P. Heinz_, Jul 18 2012
%F A108087 Sum_{k=0..n-1} A(n-k,k) = A005490(n). - _Alois P. Heinz_, Jan 05 2022
%F A108087 From _G. C. Greubel_, Dec 02 2022: (Start)
%F A108087 T(n, n) = A000012(n).
%F A108087 T(n, n-1) = A000027(n).
%F A108087 T(n, n-2) = A002522(n-1).
%F A108087 T(n, n-3) = A005491(n-2).
%F A108087 T(n, n-4) = A005492(n+1).
%F A108087 T(2*n, n) = A134980(n).
%F A108087 T(2*n, n+1) = A124824(n), n >= 1.
%F A108087 Sum_{k=0..n} T(n, k) = A347420(n). (End)
%e A108087 Array A(n,k) begins:
%e A108087    1,   1,   1,    1,    1,     1,     1,     1,     1,      1, ... A000012;
%e A108087    1,   2,   3,    4,    5,     6,     7,     8,     9,     10, ... A000027;
%e A108087    2,   5,  10,   17,   26,    37,    50,    65,    82,    101, ... A002522;
%e A108087    5,  15,  37,   77,  141,   235,   365,   537,   757,   1031, ... A005491;
%e A108087   15,  52, 151,  372,  799,  1540,  2727,  4516,  7087,  10644, ... A005492;
%e A108087   52, 203, 674, 1915, 4736, 10427, 20878, 38699, 67340, 111211, ... ;
%e A108087 Antidiagonal triangle, T(n, k), begins as:
%e A108087      1;
%e A108087      1,    1;
%e A108087      2,    2,    1;
%e A108087      5,    5,    3,    1;
%e A108087     15,   15,   10,    4,   1;
%e A108087     52,   52,   37,   17,   5,   1;
%e A108087    203,  203,  151,   77,  26,   6,  1;
%e A108087    877,  877,  674,  372, 141,  37,  7,  1;
%e A108087   4140, 4140, 3263, 1915, 799, 235, 50,  8,  1;
%p A108087 with(combinat):
%p A108087 A:= (n, k)-> add(binomial(n, i) * k^i * bell(n-i), i=0..n):
%p A108087 seq(seq(A(d-k, k), k=0..d), d=0..12);  # _Alois P. Heinz_, Jul 18 2012
%t A108087 Unprotect[Power]; 0^0 = 1; A[n_, k_] := Sum[Binomial[n, i] * k^i * BellB[n - i], {i, 0, n}]; Table[Table[A[d - k, k], {k, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Nov 05 2015, after _Alois P. Heinz_ *)
%o A108087 (PARI) f(n,k)=round (suminf(i=0,(i+k)^n/i!)/exp(1));
%o A108087 g(n,k)=for(k=0,k,print1(f(n,k),",")) \\ prints k+1 terms of n-th row
%o A108087 (Magma)
%o A108087 A108087:= func< n,k | (&+[Binomial(n-k,j)*k^j*Bell(n-k-j): j in [0..n-k]]) >;
%o A108087 [A108087(n,k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Dec 02 2022
%o A108087 (SageMath)
%o A108087 def A108087(n,k): return sum( k^j*bell_number(n-k-j)*binomial(n-k,j) for j in range(n-k+1))
%o A108087 flatten([[A108087(n,k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Dec 02 2022
%Y A108087 Cf. A000012, A000027, A000110, A002522, A005490, A005491.
%Y A108087 Cf. A005492, A005493, A005494, A045379, A086659, A124824.
%Y A108087 Main diagonal gives A134980.
%Y A108087 Antidiagonal sums give A347420.
%K A108087 nonn,tabl
%O A108087 0,4
%A A108087 _Gerald McGarvey_, Jun 05 2005