A288875 -id:A288875 - cp's OEIS frontend

A028338 Triangle of coefficients in expansion of (x+1)(x+3)...*(x + 2n - 1) in rising powers of x.

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

Offset: 0

Bill Gosper

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

			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

T. D. Noe, Rows n=0..50 of triangle, flattened
Priyavrat Deshpande, Krishna Menon, and Anurag Singh, A combinatorial statistic for labeled threshold graphs, arXiv:2103.03865 [math.CO], 2021.
Thomas Godland and Zakhar Kabluchko, Projections and angle sums of permutohedra and other polytopes, arXiv:2009.04186 [math.MG], 2020.
Thomas Godland and Zakhar Kabluchko, Projections and Angle Sums of Belt Polytopes and Permutohedra, Res. Math. (2023) Vol. 78, Art. No. 140.
Z. Kabluchko, V. Vysotsky, and 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.
Bruce E. Sagan and Joshua P. Swanson, q-Stirling numbers in type B, arXiv:2205.14078 [math.CO], 2022.

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).

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

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 *)

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
T(n, m) = (2*n-1)*T(n-1,m) + T(n-1,m-1) with T(n, 0) = (2*n-1)!! and T(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

A334670 a(n) = (2n+1)!! (Sum_{k=1..n} 1/(2*k+1)).

Original entry on oeis.org

0, 1, 8, 71, 744, 9129, 129072, 2071215, 37237680, 741975345, 16236211320, 387182170935, 9995788416600, 277792140828825, 8269430130712800, 262542617405726175, 8855805158351474400, 316285840413064454625, 11924219190760084593000, 473245342972281190686375, 19722890048636406588957000

Offset: 0

Seiichi Manyama, Sep 10 2020

nonn

			a(1) = 3 * (1/3) = 1.
a(2) = 3*5 * (1/3 + 1/5) = 8.
a(3) = 3*5*7 * (1/3 +1/5 + 1/7) = 71.

Seiichi Manyama, Table of n, a(n) for n = 0..403

Column k=1 of A335095.

Cf. A001147, A001705, A004041, A288875, A334000, A334066.

a[n_] := (2*n + 1)!! * Sum[1/(2*k + 1), {k, 1, n}]; Array[a, 21, 0] (* Amiram Eldar, Apr 29 2021 *)

{a(n) = prod(k=1, n, 2*k+1)*sum(k=1, n, 1/(2*k+1))}

{a(n) = if(n<2, n, 4*n*a(n-1)-(2*n-1)^2*a(n-2))}

a(n) + A001147(n+1) = A004041(n).
a(n) = (2*n+1) * a(n-1) + A001147(n) for n>0.
P-finite with recurrence a(n) = 4*n*a(n-1) - (2*n-1)^2 * a(n-2) for n>1.

A290595 Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0.

Original entry on oeis.org

1, 1, 2, 4, 19, 4, 28, 222, 147, 8, 280, 3194, 4128, 887, 16, 3640, 55024, 113566, 52538, 4835, 32, 58240, 1107336, 3268788, 2562676, 555684, 25167, 64, 1106560, 25526192, 100544412, 117517960, 45415640, 5301150, 128203, 128, 24344320, 663605680, 3325767376, 5352311764, 3189383200, 695714590, 47537320, 646519, 256, 608608000, 19213911360, 118361719296, 248493947496, 208996478388, 72479948400, 9696965250, 410038434, 3245139, 512

Offset: 0

Wolfdieter Lang, Aug 08 2017

The ordinary generating function (o.g.f.) of the (n+1)-th diagonal sequence of the Sheffer triangle A286718 = ((1 - 3*x)^(-1/3), -log(1 - 3*x)/3), called |S1hat[3,1]|, is GD(3,1;n,x) = P(n, x)/(1 - x)^(2*n+1), with the row polynomials P(n, x) = Sum_{k=0..n} T(n, k)*x^k, n >= 0.

For the two parameter Sheffer case |S1hat[d,a]| = ((1 - d*x)^{-a/d}, -log(1 - d*x)/d) (with gcd(d,a) = 1, d >=0, a >= 0, and for d = 1 one takes a = 0) the e.g.f. ED(t, x) of the o.g.f.s {GD(d,a;n,x)}_{n>=0} of the diagonal sequences with elements D(d,a;n,m) = |S1hat[d,a]|(n+m, m) (n=0 for the main diagonal) is of interest. It can be computed via Lagrange's theorem. For the special Sheffer case (1, f(x)) this has been done by P. Bala (see the link). This method can be generalized for Sheffer (g(x), f(x)), as shown in the W. Lang link.

			The triangle T(n, k) begins:
n\k        0        1         2         3        4       5      6   7 ...
0:         1
1:         1        2
2:         4       19         4
3:        28      222       147         8
4:       280     3194      4128       887       16
5:      3640    55024    113566     52538     4835      32
6:     58240  1107336   3268788   2562676   555684   25167      6
7:   1106560 25526192 100544412 117517960 45415640 5301150 128203 128
...
n = 8: 24344320 663605680 3325767376 5352311764 3189383200 695714590 47537320 646519 256,
n = 9: 608608000 19213911360 118361719296 248493947496 208996478388 72479948400 9696965250 410038434 3245139 512.
n = 3: The o.g.f. of the 4th diagonal sequence of A286718, [28, 418, 2485, ...] = A024213(n+1), n >= 0, is P(3, x) = (28 + 222*x + 147*x^2 + 8*x^3)/(1 - 3*x)^7.

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
Wolfdieter Lang, On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.

Cf. A024213, A286718, A288875 ([2,1] case).

T(n, k) = [x^k] P(n, x) with the numerator polynomials of the o.g.f. GD(n, x) = P(n, x)/(1-x)^(2*n+1) of the (n+1)-th diagonal sequence of the triangle A286718. See a comment above.

cp's OEIS Frontend

A028338 Triangle of coefficients in expansion of (x+1)(x+3)...*(x + 2n - 1) in rising powers of x.

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A334670 a(n) = (2n+1)!! (Sum_{k=1..n} 1/(2*k+1)).

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A290595 Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0.

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cp's OEIS Frontend

A028338 Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A334670 a(n) = (2*n+1)!! * (Sum_{k=1..n} 1/(2*k+1)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A290595 Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0.

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Author

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Comments

Examples

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Formula

A028338 Triangle of coefficients in expansion of (x+1)(x+3)...*(x + 2n - 1) in rising powers of x.

A334670 a(n) = (2n+1)!! (Sum_{k=1..n} 1/(2*k+1)).