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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

G. C. Greubel, Table of n, a(n) for the first 27 rows, flattened

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).

Cf. A000254, A165674 and A165675, A028421 and A126671.

Sequence in context: A144061 A085792 A108123 * A144252 A248033 A318254

Adjacent sequences:  A105951 A105952 A105953 * A105955 A105956 A105957

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 October 2 17:44 EDT 2022. Contains 357228 sequences. (Running on oeis4.)