0,5

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 (gerald.mcgarvey(AT)comcast.net), Aug 27 2005

A recasting of A093905 and A067176. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 01 2009]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

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.

(End)

Leroy Quet, Home Page (listed in lieu of email address)

Arthur T. Benjamin, David Gaebler and Robert Gaebler, A Combinatorial Approach to Hyperharmonic Numbers, (#A15)

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: -ln(1-x)/(1-x)^n. - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), 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 (gerald.mcgarvey(AT)comcast.net), 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 (gerald.mcgarvey(AT)comcast.net), Aug 27 2005

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

If we replace n by (n+1) in Gerard McGarvey's formulae for the row coefficients we find Wiggen's triangle A028421 and their o.g.f.s lead to Wood's polynomials A126671; see A165674.

(End)

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

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}]] (both from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 27 2005)

To view the table: Table[ a[n, m], {m, 8}, {n, 0, m + 1}] // TableForm

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+1)/(n-k), {k, 0, n-1}].) starting at n=1.

Column 3 = A001711 (Generalized Stirling numbers: a(n) = sum((-1)^(n+k)*(k+1)*3^k*stirling1(n+1, k+1), k=0..n).).

Column 4 = A001716 (Generalized Stirling numbers: a(n) = sum((-1)^(n+k)*(k+1)*4^k*stirling1(n+1, k+1), k=0..n).).

Column 5 = A001721 (Generalized Stirling numbers: a(n)=sum((-1)^(n+k)*binomial(k+1, 1)*5^k*stirling1(n+1, k+1), k=0..n).).

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).

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)

Cf. A165674 and A165675; A028421 and A126671.

(End)

Sequence in context: A144061 A085792 A108123 this_sequence A144252 A002130 A089145

Adjacent sequences: A105951 A105952 A105953 this_sequence A105955 A105956 A105957

easy,nonn,tabl

Leroy Quet Jun 26 2005

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 27 2005