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%I #56 Dec 30 2023 16:23:03
%S 1,1,1,1,3,2,1,5,11,6,1,7,26,50,24,1,9,47,154,274,120,1,11,74,342,
%T 1044,1764,720,1,13,107,638,2754,8028,13068,5040,1,15,146,1066,5944,
%U 24552,69264,109584,40320,1,17,191,1650,11274,60216,241128,663696,1026576
%N Array read by descending antidiagonals: A(n, k) = (n + 1)! * H(k, n + 1), where H(n, k) is a higher-order harmonic number, H(0, k) = 1/k and H(n, k) = Sum_{j=1..k} H(n-1, j), for 0 <= k <= n.
%C Antidiagonal sums are A093345 (n! * (1 + Sum_{i=1..n}((1/i)*Sum_{j=0..i-1} 1/j!))). - _Gerald McGarvey_, Aug 27 2005
%C A recasting of A093905 and A067176. - _R. J. Mathar_, Mar 01 2009
%C The triangular array of this sequence is the reversal of A165675 which is related to the asymptotic expansion of the higher order exponential integral E(x,m=2,n); see also A165674. - _Johannes W. Meijer_, Oct 16 2009
%H G. C. Greubel, <a href="/A105954/b105954.txt">Table of n, a(n) for the first 27 rows, flattened</a>
%H Arthur T. Benjamin, David Gaebler and Robert Gaebler, <a href="http://www.emis.de/journals/INTEGERS/papers/d15/d15.Abstract.html">A Combinatorial Approach to Hyperharmonic Numbers</a>, INTEGERS, Electronic Journal of Combinatorial Number Theory, Volum 3, #A15, 2003.
%F A(n, k) = (Harmonic(n + k) - Harmonic(k - 1))*(n + k)!/(k - 1)! if k > 0, otherwise n!.
%F From _Gerald McGarvey_, Aug 27 2005, edited by _Peter Luschny_, Jul 02 2022: (Start)
%F E.g.f. for column k: -log(1 - x)/(x*(1 - x)^k).
%F Row 3 is r(n) = 4*n^3 + 18*n^2 + 22*n + 6.
%F Row 4 is r(n) = 5*n^4 + 40*n^3 + 105*n^2 + 100*n + 24.
%F Row 5 is r(n) = 6*n^5 + 75*n^4 + 340*n^3 + 675*n^2 + 548*n + 120.
%F Row 6 is r(n) = 7*n^6 + 126*n^5 + 875*n^4 + 2940*n^3 + 4872*n^2 + 3528*n + 720.
%F Row 7 is r(n) = 8*n^7 + 196*n^6 + 1932*n^5 + 9800*n^4 + 27076*n^3 + 39396*n^2 + 26136*n + 5040.
%F The sum of the polynomial coefficients for the n-th row is |S1(n, 2)|, which are the unsigned Stirling1 numbers which appear in column 1.
%F A(m, n) = Sum_{k=1..m} n*A094645(m, n)*(n+1)^(k-1). (A094645 is Generalized Stirling number triangle of first kind, e.g.f.: (1-y)^(1-x).) (End)
%F In Gerard McGarvey's formulas for the row coefficients we find Wiggen's triangle A028421 and their o.g.f.s lead to Wood's polynomials A126671; see A165674. - _Johannes W. Meijer_, Oct 16 2009
%F A(n, k) = (n + 1)*((n + k)! / k!)*hypergeom([-n, 1, 1], [2, k + 1], 1). - _Peter Luschny_, Jul 01 2022
%e a(2, 2) = (1 + (1 + 1/2) + (1 + 1/2 + 1/3))*6 = 26.
%e Array A(n, k) begins:
%e [n\k] 0 1 2 3 4 5 6
%e -------------------------------------------------------------------
%e [0] 1, 1, 1, 1, 1, 1, 1, ...
%e [1] 1, 3, 5, 7, 9, 11, 13, ...
%e [2] 2, 11, 26, 47, 74, 107, 146, ...
%e [3] 6, 50, 154, 342, 638, 1066, 1650, ...
%e [4] 24, 274, 1044, 2754, 5944, 11274, 19524, ...
%e [5] 120, 1764, 8028, 24552, 60216, 127860, 245004, ...
%e [6] 720, 13068, 69264, 241128, 662640, 1557660, 3272688, ...
%e [7] 5040, 109584, 663696, 2592720, 7893840, 20355120, 46536624, ...
%p H := proc(n, k) option remember; if n = 0 then 1/k else add(H(n - 1, j), j = 1..k) fi end: A := (n, k) -> (n + 1)!*H(k, n + 1):
%p # Alternative with standard harmonic number:
%p A := (n, k) -> if k = 0 then n! else (harmonic(n + k) - harmonic(k - 1))*(n + k)! / (k - 1)! fi:
%p for n from 0 to 7 do seq(A(n, k), k = 0..6) od;
%p # Alternative with hypergeometric formula:
%p A := (n, k) -> (n+1)*((n + k)! / k!)*hypergeom([-n, 1, 1], [2, k+1], 1):
%p seq(print(seq(simplify(A(n, k)), k = 0..6)), n=0..7); # _Peter Luschny_, Jul 01 2022
%t H[0, m_] := 1/m; H[n_, m_] := Sum[H[n - 1, k], {k, m}]; a[n_, m_] := m!H[n, m]; Flatten[ Table[ a[i, n - i], {n, 10}, {i, n - 1, 0, -1}]]
%t Table[ a[n, m], {m, 8}, {n, 0, m + 1}] // TableForm (* to view the table *)
%t (* _Robert G. Wilson v_, Jun 27 2005 *)
%Y Column 0 = A000142 (factorial numbers).
%Y Column 1 = A000254 (Stirling numbers of first kind s(n, 2)) starting at n=1.
%Y Column 2 = A001705 (Generalized Stirling numbers: a(n) = n!*Sum_{k=0..n-1}(k+1)/(n-k)), starting at n=1.
%Y Column 3 = A001711 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*(k+1)*3^k*stirling1(n+1, k+1)).
%Y Column 4 = A001716 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*(k+1)*4^k*stirling1(n+1, k+1)).
%Y Column 5 = A001721 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*binomial(k+1, 1)*5^k*stirling1(n+1, k+1)).
%Y Column 6 = A051524 (2nd unsigned column of A051338) starting at n=1.
%Y Column 7 = A051545 (2nd unsigned column of A051339) starting at n=1.
%Y Column 8 = A051560 (2nd unsigned column of A051379) starting at n=1.
%Y Column 9 = A051562 (2nd unsigned column of A051380) starting at n=1.
%Y Column 10= A051564 (2nd unsigned column of A051523) starting at n=1.
%Y 2nd row is A005408 (2n - 1, starting at n=1).
%Y 3rd row is A080663 (3n^2 - 1, starting at n=1).
%Y Cf. A000254, A165674 and A165675, A028421 and A126671.
%K nonn,tabl,easy
%O 0,5
%A _Leroy Quet_, Jun 26 2005
%E More terms from _Robert G. Wilson v_, Jun 27 2005
%E Edited by _Peter Luschny_, Jul 02 2022
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