A293318 -id:A293318 - 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

A109692 Triangle of coefficients in expansion of (1+x)(1+3x)(1+5x)(1+7x)...*(1+(2n-1)x).

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 9, 23, 15, 1, 16, 86, 176, 105, 1, 25, 230, 950, 1689, 945, 1, 36, 505, 3480, 12139, 19524, 10395, 1, 49, 973, 10045, 57379, 177331, 264207, 135135, 1, 64, 1708, 24640, 208054, 1038016, 2924172, 4098240, 2027025

Offset: 0

Philippe Deléham, Aug 08 2005

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 2, 3, 4, 5, 6, 7, 8, 9, ...] where DELTA is the operator defined in A084938.

T(n,k), 0 <= k <= n, is the number of elements in the Coxeter group B_n with absolute length k. - Jose Bastidas, Jul 14 2023

			Triangle T(n,k) begins:
  1;
  1,  1;
  1,  4,   3;
  1,  9,  23,   15;
  1, 16,  86,  176,   105;
  1, 25, 230,  950,  1689,   945;
  1, 36, 505, 3480, 12139, 19524, 10395;
  ...

Seiichi Manyama, Rows n = 0..139, flattened

Cf. A039758 (signed version). A028338 transposed.
Diagonals: A000012, A000290, A024196, A024197, A024198; A001147, A004041, A028339, A028340, A028341.
Row sums: A000165.
Central terms: A293318.
Cf. A161198 (transposed scaled triangle version).

nmax:=8; mmax:=nmax: for n from 0 to nmax do a(n, n) := doublefactorial(2*n-1) od: for n from 0 to nmax do a(n, 0):=1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := a(n-1,m) + (2*n-1)*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,m) = T(n-1,m) + (2*n-1)*T(n-1,m-1) with T(n,n) = (2*n-1)!! and T(n,0) = 1. - Johannes W. Meijer, Jun 08 2009

A348085 a(n) = [x^n] Product_{k=1..2n} 1/(1 - (2k-1) * x).

Original entry on oeis.org

1, 4, 170, 13776, 1652442, 262842580, 52116296024, 12380577235040, 3427841258566890, 1083931844930932140, 385417972804020879450, 152219732613102667656000, 66113646914860527721527960, 31319437721634527178263452656

Offset: 0

Seiichi Manyama, Sep 28 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..277

Cf. A039755, A293318, A348082, A348084, A348087.

a(n) = polcoef(1/prod(k=1, 2*n, 1-(2*k-1)*x+x*O(x^n)), n);

a(n) = if(n==0, 1, -sum(k=0, 2*n-1, (-1)^k*(2*k+1)^(3*n-1)*binomial(2*n-1, k))/(2^(2*n-1)*(2*n-1)!));

a(n) = A039755(3*n-1,2*n-1) for n > 0.
a(n) = (-1/(2^(2*n-1) * (2*n-1)!)) * Sum_{k=0..2*n-1} (-1)^k * (2*k+1)^(3*n-1) * binomial(2*n-1,k) for n > 0.
a(n) ~ 3^(3*n - 1/2) * n^(n - 1/2) / (sqrt(2*Pi*(1-c)) * (3 - 2*c)^n * c^(2*n - 1/2) * exp(n)), where c = -LambertW(-3*exp(-3/2)/2) = 0.62578253420128292... - Vaclav Kotesovec, Oct 02 2021
From Seiichi Manyama, May 16 2025: (Start)
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-1,k+2*n-1) * Stirling2(k+2*n-1,2*n-1) for n > 0.
a(n) = Sum_{k=0..n} (-2)^k * (4*n-1)^(n-k) * binomial(3*n-1,k+2*n-1) * Stirling2(k+2*n-1,2*n-1) for n > 0. (End)

A346543 a(n) = [x^n] Product_{k=1..2n} (x + (2k-1)^2).

Original entry on oeis.org

1, 10, 1974, 1234948, 1601489318, 3541644282540, 11934462103156540, 56947950742822581960, 365458809637016986262790, 3035813466162156094097686300, 31694033885101849517370941522644, 406222401519003083851664224927890360, 6271146756206887832796744632163811733084

Offset: 0

Seiichi Manyama, Sep 27 2021

nonn

			(1/3!) * (arcsin(x))^3 = x^3/3! + 10 * x^5/5! + ... . So a(1) =10.
(1/5!) * (arcsin(x))^5 = x^5/5! + 35 * x^7/7! + 1974 * x^9/9! + ... . So a(2) = 1974.

Cf. A008956, A016754, A234324, A293318.

Table[SeriesCoefficient[Product[(x + (2*k-1)^2), {k, 1, 2*n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2021 *)

a(n) = polcoef(prod(k=1, 2*n, x+(2*k-1)^2), n);

a(n) = A008956(2*n,n).
a(n) = (4*n+1)! * [x^(4*n+1)] (1/(2*n+1)!) * (arcsin(x))^(2*n+1).
a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 121.8904568356133798202328777176879971969471503678428704459083316116687149... and c = 0.1081647814943965981694666415038643176470488612855594762896553127... - Vaclav Kotesovec, Oct 16 2021

A383704 a(n) = [x^n] Product_{k=0..2n-1} (x - (-1)^k (2*k+1)).

Original entry on oeis.org

1, 2, -34, -540, 26614, 805980, -66399124, -2972817848, 343902030758, 20389669252524, -3039312653124540, -224361715353976200, 40941662601331486396, 3617518823154571788440, -781104190733806836937320, -80375840650247250199417200, 20044038897159722534821833990

Offset: 0

Seiichi Manyama, May 06 2025

sign

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A293318.

[&+[((4*n-1)^k * 4^(n-k) * Binomial(n+k,n) * StirlingFirst(2*n,n+k)): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, May 07 2025

Table[Sum[(4*n-1)^k*4^(n-k)*Binomial[n+k,n]*StirlingS1[2*n,n+k],{k,0,n}],{n,0,17}] (* Vincenzo Librandi, May 07 2025 *)

a(n) = polcoef(prod(k=0, 2*n-1, x-(-1)^k*(2*k+1)), n);

a(n) = Sum_{k=0..n} (-(4*n-3))^k * 4^(n-k) * binomial(n+k,n) * |Stirling1(2*n,n+k)|.
a(n) = Sum_{k=0..n} (4*n-1)^k * 4^(n-k) * binomial(n+k,n) * Stirling1(2*n,n+k).
a(n) = (2*n)! * [x^(2*n)] 1/f(x)^(4*n-3) * log(f(x))^n / n!, where f(x) = 1/(1 - 4*x)^(1/4).

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

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A109692 Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).

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A348085 a(n) = [x^n] Product_{k=1..2*n} 1/(1 - (2*k-1) * x).

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A346543 a(n) = [x^n] Product_{k=1..2*n} (x + (2*k-1)^2).

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A383704 a(n) = [x^n] Product_{k=0..2*n-1} (x - (-1)^k * (2*k+1)).

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

A109692 Triangle of coefficients in expansion of (1+x)(1+3x)(1+5x)(1+7x)...*(1+(2n-1)x).

A348085 a(n) = [x^n] Product_{k=1..2n} 1/(1 - (2k-1) * x).

A346543 a(n) = [x^n] Product_{k=1..2n} (x + (2k-1)^2).

A383704 a(n) = [x^n] Product_{k=0..2n-1} (x - (-1)^k (2*k+1)).