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A028338 Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x. 27
1, 1, 1, 3, 4, 1, 15, 23, 9, 1, 105, 176, 86, 16, 1, 945, 1689, 950, 230, 25, 1, 10395, 19524, 12139, 3480, 505, 36, 1, 135135, 264207, 177331, 57379, 10045, 973, 49, 1, 2027025, 4098240, 2924172, 1038016, 208054, 24640, 1708, 64, 1, 34459425, 71697105, 53809164, 20570444, 4574934, 626934, 53676, 2796, 81, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Exponential Riordan array (1/sqrt(1-2*x), log(1/sqrt(1-2*x))). - Paul Barry, May 09 2011

The o.g.f.s D(d, x) of the column sequences, for d, d >= 0,(d=0 for the main diagonal) are P(d, x)/(1 - x)^(2*d+1), with the row polynomial P(d, x) = Sum_{m=0..d} A288875(d, m)*x^m. See A288875 for details. - Wolfdieter Lang, Jul 21 2017

LINKS

T. D. Noe, Rows n=0..50 of triangle, flattened

Z. Kabluchko, V. Vysotsky, D. Zaporozhets, Convex hulls of random walks, hyperplane arrangements, and Weyl chambers, arXiv preprint arXiv:1510.04073 [math.PR], 2015.

Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.

FORMULA

Triangle T(n, k), read by rows, given by [1, 2, 3, 4, 5, 6, 7, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 20 2005

T(n, k) = Sum_{i=k..n} (-2)^(n-i) * binomial(i, k) * s(n, i) where s(n, k) are signed Stirling numbers of the first kind. - Francis Woodhouse (fwoodhouse(AT)gmail.com), Nov 18 2005

G.f. of row polynomials in y: 1/(1-(x+x*y)/(1-2*x/(1-(3*x+x*y)/(1-4*x/(1-(5*x+x*y)/(1-6*x*y/(1-... (continued fraction). - Paul Barry, Feb 07 2009

a(n, m) = (2*n-1)*a(n-1,m) + a(n-1,m-1) with a(n, 0) = (2*n-1)!! and a(n, n) = 1. - Johannes W. Meijer, Jun 08 2009

From Wolfdieter Lang, May 09 2017: (Start)

E.g.f. of row polynomials in y: (1/sqrt(1-2*x))*exp(-y*log(sqrt(1-2*x))) = exp(-(1+y)*log(sqrt(1-2*x))) = 1/sqrt(1-2*x)^(1+y).

E.g.f. of column m sequence: (1/sqrt(1-2*x))* (-log(sqrt(1-2*x)))^m/m!. For the special Sheffer, also known as exponential Riordan array, see a comment above. (End)

Boas-Buck type recurrence for column sequence k: T(n, k) = (n!/(n - k)) * Sum_{p=k..n-1} 2^(n-1-p)*(1 + 2*k*beta(n-1-p))*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, and beta(k) = A002208(k+1)/A002209(k+1). See a comment and references in A286718. - Wolfdieter Lang, Aug 09 2017

EXAMPLE

G.f. for n = 4: (x + 1)*(x + 3)*(x + 5)*(x + 7) = 105 + 176*x + 86*x^2 + 16*x^3 + x^4.

The triangle T(n, k) begins:

n\k       0        1        2        3       4      5     6    7  8  9

0:        1

1:        1        1

2:        3        4        1

3:       15       23        9        1

4:      105      176       86       16       1

5:      945     1689      950      230      25      1

6:    10395    19524    12139     3480     505     36     1

7:   135135   264207   177331    57379   10045    973    49    1

8:  2027025  4098240  2924172  1038016  208054  24640  1708   64  1

9: 34459425 71697105 53809164 20570444 4574934 626934 53676 2796 81  1

...

row n = 10: 654729075 1396704420 1094071221 444647600 107494190 16486680 1646778 106800 4335 100 1.

...  reformatted and extended. - Wolfdieter Lang, May 09 2017

O.g.f.s of diagonals d >= 0: D(2, x) = (3 + 8*x + x^2)/(1 - x)^5 generating [3, 23, 86, ...] = A024196(n+1), from the row d=2 entries of A288875 [3, 8, 1]. - Wolfdieter Lang, Jul 21 2017

Boas-Buck recurrence for column k=2 and n=4: T(4, 2) = (4!/2)*(2*(1+4*(5/12 )*T(2, 2)/2! + 1*(1 + 4*(1/2))*T(3,2)/3!) = (4!/2)*(8/3*1 + 3*9/3!) = 86. - Wolfdieter Lang, Aug 11 2017

MAPLE

nmax:=8; for n from 0 to nmax do a(n, 0) := doublefactorial(2*n-1) od: for n from 0 to nmax do a(n, n) := 1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := (2*n-1)*a(n-1, m) + a(n-1, m-1) od; od: seq(seq(a(n, m), m=0..n), n=0..nmax); # Johannes W. Meijer, Jun 08 2009, revised Nov 25 2012

MATHEMATICA

T[n_, k_] := Sum[(-2)^(n-i) Binomial[i, k] StirlingS1[n, i], {i, k, n}] (* Woodhouse *)

Join[{1}, Flatten[Table[CoefficientList[Expand[Times@@Table[x+i, {i, 1, 2n+1, 2}]], x], {n, 0, 10}]]] (* Harvey P. Dale, Jan 29 2013 *)

CROSSREFS

A039757 is signed version.

Row sums: A000165.

Columns: A001147, A004041, A028339, A028340, A028341; A000012, A000290, A024196, A024197, A024198.

Diagonals: A000012, A000290(n+1), A024196(n+1), A024197(n+1), A024198(n+1).

A161198 is a scaled triangle version and A109692 is a transposed triangle version.

Central terms: A293318.

Cf. A286718, A002208(n+1)/A002209(n+1).

Sequence in context: A059110 A100326 * A039757 A136228 A154829 A215241

Adjacent sequences:  A028335 A028336 A028337 * A028339 A028340 A028341

KEYWORD

tabl,nonn,easy,nice

AUTHOR

Bill Gosper

STATUS

approved

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Last modified February 24 11:12 EST 2018. Contains 299603 sequences. (Running on oeis4.)