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A105954 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. 9
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, 1044, 1764, 720, 1, 13, 107, 638, 2754, 8028, 13068, 5040, 1, 15, 146, 1066, 5944, 24552, 69264, 109584, 40320, 1, 17, 191, 1650, 11274, 60216, 241128, 663696, 1026576 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Antidiagonal sums are A093345 (n! * (1 + Sum_{i=1..n}((1/i)*Sum_{j=0..i-1} 1/j!))). - Gerald McGarvey, Aug 27 2005
A recasting of A093905 and A067176. - R. J. Mathar, Mar 01 2009
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
LINKS
Arthur T. Benjamin, David Gaebler and Robert Gaebler, A Combinatorial Approach to Hyperharmonic Numbers, INTEGERS, Electronic Journal of Combinatorial Number Theory, Volum 3, #A15, 2003.
FORMULA
A(n, k) = (Harmonic(n + k) - Harmonic(k - 1))*(n + k)!/(k - 1)! if k > 0, otherwise n!.
From Gerald McGarvey, Aug 27 2005, edited by Peter Luschny, Jul 02 2022: (Start)
E.g.f. for column k: -log(1 - x)/(x*(1 - x)^k).
Row 3 is r(n) = 4*n^3 + 18*n^2 + 22*n + 6.
Row 4 is r(n) = 5*n^4 + 40*n^3 + 105*n^2 + 100*n + 24.
Row 5 is r(n) = 6*n^5 + 75*n^4 + 340*n^3 + 675*n^2 + 548*n + 120.
Row 6 is r(n) = 7*n^6 + 126*n^5 + 875*n^4 + 2940*n^3 + 4872*n^2 + 3528*n + 720.
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.
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.
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)
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
A(n, k) = (n + 1)*((n + k)! / k!)*hypergeom([-n, 1, 1], [2, k + 1], 1). - Peter Luschny, Jul 01 2022
EXAMPLE
a(2, 2) = (1 + (1 + 1/2) + (1 + 1/2 + 1/3))*6 = 26.
Array A(n, k) begins:
[n\k] 0 1 2 3 4 5 6
-------------------------------------------------------------------
[0] 1, 1, 1, 1, 1, 1, 1, ...
[1] 1, 3, 5, 7, 9, 11, 13, ...
[2] 2, 11, 26, 47, 74, 107, 146, ...
[3] 6, 50, 154, 342, 638, 1066, 1650, ...
[4] 24, 274, 1044, 2754, 5944, 11274, 19524, ...
[5] 120, 1764, 8028, 24552, 60216, 127860, 245004, ...
[6] 720, 13068, 69264, 241128, 662640, 1557660, 3272688, ...
[7] 5040, 109584, 663696, 2592720, 7893840, 20355120, 46536624, ...
MAPLE
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):
# Alternative with standard harmonic number:
A := (n, k) -> if k = 0 then n! else (harmonic(n + k) - harmonic(k - 1))*(n + k)! / (k - 1)! fi:
for n from 0 to 7 do seq(A(n, k), k = 0..6) od;
# Alternative with hypergeometric formula:
A := (n, k) -> (n+1)*((n + k)! / k!)*hypergeom([-n, 1, 1], [2, k+1], 1):
seq(print(seq(simplify(A(n, k)), k = 0..6)), n=0..7); # Peter Luschny, Jul 01 2022
MATHEMATICA
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}]]
Table[ a[n, m], {m, 8}, {n, 0, m + 1}] // TableForm (* to view the table *)
(* Robert G. Wilson v, Jun 27 2005 *)
CROSSREFS
Column 0 = A000142 (factorial numbers).
Column 1 = A000254 (Stirling numbers of first kind s(n, 2)) starting at n=1.
Column 2 = A001705 (Generalized Stirling numbers: a(n) = n!*Sum_{k=0..n-1}(k+1)/(n-k)), starting at n=1.
Column 3 = A001711 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*(k+1)*3^k*stirling1(n+1, k+1)).
Column 4 = A001716 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*(k+1)*4^k*stirling1(n+1, k+1)).
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)).
Column 6 = A051524 (2nd unsigned column of A051338) starting at n=1.
Column 7 = A051545 (2nd unsigned column of A051339) starting at n=1.
Column 8 = A051560 (2nd unsigned column of A051379) starting at n=1.
Column 9 = A051562 (2nd unsigned column of A051380) starting at n=1.
Column 10= A051564 (2nd unsigned column of A051523) starting at n=1.
2nd row is A005408 (2n - 1, starting at n=1).
3rd row is A080663 (3n^2 - 1, starting at n=1).
Sequence in context: A144061 A085792 A108123 * A144252 A248033 A318254
KEYWORD
nonn,tabl,easy
AUTHOR
Leroy Quet, Jun 26 2005
EXTENSIONS
More terms from Robert G. Wilson v, Jun 27 2005
Edited by Peter Luschny, Jul 02 2022
STATUS
approved

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Last modified March 29 09:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)