The On-Line Encyclopedia of Integer Sequences (OEIS)

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A126671

Triangle read by rows: row n (n>=0) has g.f. Sum_{i=1..n} n!*x^i*(1+x)^(n-i)/(n+1-i).
(history; published version)

#5 by Russ Cox at Fri Mar 30 16:50:37 EDT 2012

AUTHOR

_N. J. A. Sloane (njas(AT)research.att.com) _ and Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007

Discussion

Fri Mar 30

16:50

OEIS Server: https://oeis.org/edit/global/110

#4 by N. J. A. Sloane at Thu Nov 11 07:34:06 EST 2010

LINKS

N. J. A. Sloane, <a href="/A126671/a126671.txt">Notes on Carlo Wood's Polynomials</a>

KEYWORD

nonn,tabl,new

#3 by N. J. A. Sloane at Tue Jun 01 03:00:00 EDT 2010

COMMENTS

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009: (Start)

The coefficients of the A165674 triangle are generated by the asymptotic expansion of the higher order exponential integral E(x,m=2,n). The a(n) formulae for the coefficients in the right hand columns of this triangle lead to Wiggen's triangle A028421 and their o.g.f.s. lead to the sequence given above. Some right hand columns of the A165674 triangle are A080663, A165676, A165677, A165678 and A165679.

(End)

FORMULA

E.g.f.: x*ln(1-(1+x)*y)/(x*y-1)/(1+x). - Vladeta Jovovic (vladeta(AT)Euneteunet.yurs), Feb 13 2007

CROSSREFS

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009: (Start)

Cf. A165674, A028421, A080663, A165676, A165677, A165678 and A165679.

(End)

KEYWORD

nonn,tabl,new

#2 by N. J. A. Sloane at Fri Feb 27 03:00:00 EST 2009

LINKS

N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a126671.txt">Notes on Carlo Wood's Polynomials</a>

KEYWORD

nonn,tabl,new

AUTHOR

N. J. A. Sloane (njas (AT)research.att.com) and Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007

#1 by N. J. A. Sloane at Fri May 11 03:00:00 EDT 2007

NAME

Triangle read by rows: row n (n>=0) has g.f. Sum_{i=1..n} n!*x^i*(1+x)^(n-i)/(n+1-i).

DATA

0, 0, 1, 0, 1, 3, 0, 2, 7, 11, 0, 6, 26, 46, 50, 0, 24, 126, 274, 326, 274, 0, 120, 744, 1956, 2844, 2556, 1764, 0, 720, 5160, 16008, 28092, 30708, 22212, 13068, 0, 5040, 41040, 147120, 304464, 401136, 351504, 212976, 109584, 0, 40320

OFFSET

1,6

COMMENTS

The first nonzero column gives the factorial numbers, which are Stirling_1(*,1), the rightmost diagonal gives Stirling_1(*,2), so this triangle may be regarded as interpolating between the first two columns of the Stirling numbers of the first kind.

This is a slice (the right-hand wall) through the infinite square pyramid described in the link. The other three walls give A007318 and A008276 (twice).

LINKS

N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a126671.txt">Notes on Carlo Wood's Polynomials</a>

FORMULA

Recurrence: T(n,0) = 0; for n>=0, i>=1, T(n+1,i) = (n+1)*T(n,i) + n!*binomial(n,i).

E.g.f.: x*ln(1-(1+x)*y)/(x*y-1)/(1+x). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 13 2007

EXAMPLE

Triangle begins:

0,

0, 1,

0, 1, 3,

0, 2, 7, 11,

0, 6, 26, 46, 50,

0, 24, 126, 274, 326, 274,

0, 120, 744, 1956, 2844, 2556, 1764,

0, 720, 5160, 16008, 28092, 30708, 22212, 13068,

0, 5040, 41040, 147120, 304464, 401136, 351504, 212976, 109584,

0, 40320, 367920, 1498320, 3582000, 5562576, 5868144, 4292496, 2239344, 1026576, ...

MAPLE

for n from 1 to 15 do t1:=add( n!*x^i*(1+x)^(n-i)/(n+1-i), i=1..n); series(t1, x, 100); lprint(seriestolist(%)); od:

CROSSREFS

Columns give A000142, A108217, A126672; diagonals give A000254, A067318, A126673. Row sums give A126674. Alternating row sums give A000142.

See A126682 for the full pyramid of coefficients of the underlying polynomials.

KEYWORD

nonn,tabl

AUTHOR

njas and Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007

STATUS

approved