A024197 -id:A024197 - 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

A161198 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).

Original entry on oeis.org

1, 1, 2, 3, 8, 4, 15, 46, 36, 8, 105, 352, 344, 128, 16, 945, 3378, 3800, 1840, 400, 32, 10395, 39048, 48556, 27840, 8080, 1152, 64, 135135, 528414, 709324, 459032, 160720, 31136, 3136, 128

Offset: 0

Johannes W. Meijer, Jun 08 2009, Jul 22 2011

The series expansion of (1-x)^((-1-2*n)/2) = sum(b(p)*x^p, p=0..infinity) for n = 0, 1, 2, .. can be described with b(p) = (F(p,n)/ (2*n-1)!!)*(binomial(2*p,p)/4^(p)) with F(x,n) = 2^n * product( x+(2*k-1)/2, k=1..n). The roots of the F(x,n) polynomials can be found at p = (1-2*k)/2 with k from 1 to n for n = 0, 1, 2, .. . The coefficients of the F(x,n) polynomials lead to the triangle given above. The triangle row sums lead to A001147.
Quite surprisingly we discovered that sum(b(p)*x^p, p=0..infinity) = (1-x)^(-1-2*n)/2, for n = -1, -2, .. . We assume that if m = n+1 then the value returned for product(f(k), k = m..n) is 1 and if m> n+1 then 1/product(f(k), k=n+1..m-1) is the value returned. Furthermore (1-2*n)!! = (-1)^(n+1)/(2*n-3)!! for n = 1, 2, 3 .. . This leads to b(p) = ((-1-2*n)!!/ G(p,n))*(binomial(2*p,p) /4^(p)) for n = -1, -2, .. . For the G(p,n) polynomials we found that G(p,n) = F(-p,-n). The roots of the G(p,n) polynomials can be found at p=(2*k-1)/2 with k from 1 to (-n) for n = -1, -2, .. . The coefficients of the G(p,n) polynomials lead to a second triangle that stands with its head on top of the first one. It is remarkable that the row sums lead once again to A001147.
These two triangles together look like an hourglass so we propose to call the F(p,n) and the G(p,n) polynomials the hourglass polynomials.
Triangle T(n,k), read by rows, given by (1, 2, 3, 4, 5, 6, 7, 8, 9, ...) DELTA (2, 0, 2, 0, 2, 0, 2, 0, 2, ...) where DELTA is the operator defined in A084938. Philippe Deléham, May 14 2015.

			From _Gary W. Adamson_, Jul 19 2011: (Start)
The first few rows of matrix M are:
  1, 2,  0,  0, 0, ...
  1, 3,  2,  0, 0, ...
  1, 4,  5,  2, 0, ...
  1, 5,  9,  7, 2, ...
  1, 6, 14, 16, 9, ... (End)
The first few G(p,n) polynomials are:
  G(p,-3) = 15 - 46*p + 36*p^2 - 8*p^3
  G(p,-2) = 3 - 8*p + 4*p^2
  G(p,-1) = 1 - 2*p
The first few F(p,n) polynomials are:
  F(p,0) = 1
  F(p,1) = 1 + 2*p
  F(p,2) = 3 + 8*p + 4*p^2
  F(p,3) = 15 + 46*p + 36*p^2 + 8*p^3
The first few rows of the upper and lower hourglass triangles are:
  [15, -46, 36, -8]
  [3, -8, 4]
  [1, -2]
  [1]
  [1, 2]
  [3, 8, 4]
  [15, 46, 36, 8]

Cf. A001790 [(1-x)^(-1/2)], A001803 [(1-x)^(-3/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/2)].
Cf. A002596 [(1-x)^(1/2)], A161200 [(1-x)^(3/2)] and A161202 [(1-x)^(5/2)].
A046161 gives the denominators of the series expansions of all (1-x)^((-1-2*n)/2).
A028338 is a scaled triangle version, A039757 is a scaled signed triangle version and A109692 is a transposed scaled triangle version.
A001147 is the first left hand column and equals the row sums.
A004041 is the second left hand column divided by 2, A028339 is the third left hand column divided by 4, A028340 is the fourth left hand column divided by 8, A028341 is the fifth left hand column divided by 16.
A000012, A000290, A024196, A024197 and A024198 are the first (n-m=0), second (n-m=1), third (n-m=2), fourth (n-m=3) and fifth (n-m=4) right hand columns divided by 2^m.
A074599 * A025549 is not always equals the second left hand column.
Cf. A029635. [Gary W. Adamson, Jul 19 2011]

nmax:=7; for n from 0 to nmax do a(n,n):=2^n: a(n,0):=doublefactorial(2*n-1) od: for n from 2 to nmax do for m from 1 to n-1 do a(n,m) := 2*a(n-1,m-1)+(2*n-1)*a(n-1,m) od: od: seq(seq(a(n,k), k=0..n), n=0..nmax);
nmax:=7: M := Matrix(1..nmax+1,1..nmax+1): A029635 := proc(n,k): binomial(n,k) + binomial(n-1,k-1) end: for i from 1 to nmax do for j from 1 to i+1 do M[i,j] := A029635(i,j-1) od: od: for n from 0 to nmax do B := M^n: for m from 0 to n do a(n,m):= B[1,m+1] od: od: seq(seq(a(n,m), m=0..n), n=0..nmax);
A161198 := proc(n,k) option remember; if k > n or k < 0 then 0 elif n = 0 and k = 0 then 1 else 2*A161198(n-1, k-1) + (2*n-1)*A161198(n-1, k) fi end:
seq(print(seq(A161198(n,k), k = 0..n)), n = 0..6);  # Peter Luschny, May 09 2013

nmax = 7; a[n_, 0] := (2*n-1)!!; a[n_, n_] := 2^n; a[n_, m_] := a[n, m] = 2*a[n-1, m-1]+(2*n-1)*a[n-1, m]; Table[a[n, m], {n, 0, nmax}, {m, 0, n}] // Flatten (* Jean-François Alcover, Feb 25 2014, after Maple *)

for(n=0,9, print(Vec(Ser( 2^n*prod( k=1,n, x+(2*k-1)/2 ),,n+1))))  \\ M. F. Hasler, Jul 23 2011

@CachedFunction
def A161198(n,k):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return 2*A161198(n-1,k-1)+(2*n-1)*A161198(n-1,k)
for n in (0..6): [A161198(n,k) for k in (0..n)]  # Peter Luschny, May 09 2013

a(n,m) := coeff(2^(n)*product((x+(2*k-1)/2),k=1..n), x, m) for n = 0, 1, .. ; m = 0, 1, .. .
a(n, m) = 2*a(n-1,m-1)+(2*n-1)*a(n-1,m) with a(n, n) = 2^n and a(n, 0) = (2*n-1)!!.
a(n,m) = the (m+1)-th term in the top row of M^n, where M is an infinite square production matrix; M[i,j] = A029635(i,j-1) = binomial(i, j-1) + binomial(i-1, j-2) with A029635 the (1.2)-Pascal triangle, see the examples and second Maple program. [Gary W. Adamson, Jul 19 2011]
T(n,k) = 2^k * A028338(n,k). - Philippe Deléham, May 14 2015

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

A024202 a(n) = [ (3rd elementary symmetric function of S(n))/(first elementary symmetric function of S(n)) ], where S(n) = {first n+2 odd positive integers}.

Original entry on oeis.org

1, 11, 38, 96, 205, 385, 662, 1068, 1635, 2401, 3410, 4706, 6339, 8365, 10840, 13826, 17391, 21603, 26536, 32270, 38885, 46467, 55108, 64900, 75941, 88335, 102186, 117604, 134705, 153605, 174426, 197296, 222343, 249701, 279510, 311910, 347047, 385073

Offset: 1

Clark Kimberling

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A024197.

f:= proc(n)
  if n mod 3 = 1 then (n^4+4*n^3+4*n^2+n-4)/6
  else n*(n+1)*(n^2+3*n+1)/6
  fi
end proc:
map(f, [$1..100]); # Robert Israel, Dec 30 2016

Table[Floor[n*(n + 1)*(n^2 + 3*n + 1)/6], {n, 1, 50}] (* G. C. Greubel, Dec 30 2016 *)

for(n=1,25, print1(floor(n*(n+1)*(n^2+3*n+1)/6), ", ")) \\ G. C. Greubel, Dec 30 2016

Empirical g.f.: x*(x^4-5*x^3-7*x-1) / ((x-1)^5*(x^2+x+1)). - Colin Barker, Aug 15 2014
From Robert Israel, Dec 30 2016: (Start)
a(n) = floor(A024197(n)/(n+2)^2) = floor(n*(n+1)*(n^2+3*n+1)/6).
a(n) = (n^4+4*n^3+4*n^2+n-4)/6 if n == 1 (mod 3).
Otherwise a(n) = n*(n+1)*(n^2+3*n+1)/6.
The empirical g.f. can be obtained from this. (End)

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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A161198 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).

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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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A024202 a(n) = [ (3rd elementary symmetric function of S(n))/(first elementary symmetric function of S(n)) ], where S(n) = {first n+2 odd positive integers}.

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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).