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A105954 Array read by antidiagonals: a(m,n) = m!*H(n,m), where H(n,m) is a higher-order harmonic number (H(0,m) = 1/m; H(n,m) = Sum_{k=1..m} H(n-1,k)). 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

In the array, the m index runs from 1 on, n index runs from 0 on.

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(m, n) = (H_{m+n-1} - H_{n-1})(m+n-1)!/(n-1)!, where H_k = H(1, k), a standard harmonic number. Array is read off by diagonals.

E.g.f. for column n: -log(1-x)/(1-x)^n. - Gerald McGarvey, Aug 27 2005

4th row is 4n^3 + 6n^2 - 2n - 2. 5th row is 5n^4 + 20n^3 + 15n^2 - 10n - 6. 6th row is 6n^5 + 45n^4 + 100n^3 + 45n^2 - 52n - 24. 7th row is 7n^6 + 84n^5 + 350n^4 + 560n^3 + 147n^2 - 308n - 120. 8th row is 8n^7 + 140n^6 + 924n^5 + 2800n^4 + 3556n^3 + 420n^2 - 2088n - 720. The sum of the polynomial coefficients for the m-th row is (m-1)!. A005564 begins as 6, 20, 45, 84, 140, ... - Gerald McGarvey, Aug 27 2005

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).) - Gerald McGarvey, Aug 27 2005

If we replace n with (n+1) 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

EXAMPLE

a(2,3) = (1 + (1 + 1/2) + (1 + 1/2 + 1/3))*6 = 26.

Array begins:

1    1    1    1    1     1     1     1     1 ...

1    3    5    7    9    11    13    15    17 ...

2   11   26   47   74   107   146   191   242 ...

6   50  154  342  638  1066  1650  2414  3382 ...

24 274 1044 2754 5944 11274 19524 31594 48504 ...

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

Cf. A000254.

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 (second unsigned column of triangle A051338) starting at n=1.

Column 7 = A051545 (second unsigned column of triangle A051339) starting at n=1.

Column 8 = A051560 (second unsigned column of triangle A051379) starting at n=1.

Column 9 = A051562 (second unsigned column of triangle A051380) starting at n=1.

Column 10= A051564 (second unsigned column of triangle 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. A165674 and A165675; A028421 and A126671. - Johannes W. Meijer, Oct 16 2009

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

STATUS

approved

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Last modified December 5 09:45 EST 2021. Contains 349543 sequences. (Running on oeis4.)