A028338
Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
Original entry on oeis.org
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
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.
A161198 is a scaled triangle version and
A109692 is a transposed triangle version.
-
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 *)
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
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]
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.
-
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
A004041
Scaled sums of odd reciprocals: a(n) = (2*n + 1)!!*(Sum_{k=0..n} 1/(2*k + 1)).
Original entry on oeis.org
1, 4, 23, 176, 1689, 19524, 264207, 4098240, 71697105, 1396704420, 29985521895, 703416314160, 17901641997225, 491250187505700, 14459713484342175, 454441401368236800, 15188465029114325025, 537928935889764226500
Offset: 0
Joe Keane (jgk(AT)jgk.org)
(arctanh(x))^2 = x^2 + 2/3*x^4 + 23/45*x^6 + 44/105*x^8 + ...
Equals second left hand column of
A028338 triangle.
Equals second right hand column of
A109692 triangle.
Equals second left hand column of
A161198 triangle divided by 2.
(End)
-
Table[(-1)^(n + 1)* Sum[(-2)^(n - k) k (-1)^(n - k) StirlingS1[n + 1, k + 1], {k, 0, n}], {n, 1, 18}] (* Zerinvary Lajos, Jul 08 2009 *)
FunctionExpand@Table[(2 n + 1)!! (Log[4] + HarmonicNumber[n + 1/2])/2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 13 2016 *)
A039757
Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...*(x-(2*n-1)).
Original entry on oeis.org
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
Ruedi Suter (suter(AT)math.ethz.ch)
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
- Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
- Priyavrat Deshpande, Krishna Menon, and Anurag Singh, Counting regions of the boxed threshold arrangement, arXiv:2101.12060 [math.CO], 2021.
- Ruedi Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.
- Z. Kabluchko, V. Vysotsky, and D. Zaporozhets, Convex hulls of random walks, hyperplane arrangements, and Weyl chambers, arXiv preprint arXiv:1510.04073 [math.PR], 2015-2017.
A161198 is an unsigned scaled triangle version.
A109692 is an unsigned transposed triangle version.
A000165(n)*(-1)^n (alternating row sums).
-
nmax:=8; mmax:=nmax: for n from 0 to nmax do a(n, 0) := (-1)^n*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) := a(n-1, m-1)-(2*n-1)*a(n-1, m) od; od: seq(seq(a(n, m), m=0..n), n=0..nmax); # Johannes W. Meijer, Jun 08 2009, revised Nov 29 2012
-
a[n_, m_] := a[n, m] = a[n-1, m-1] - (2*n-1)*a[n-1, m]; a[n_, 0] := (-1)^n*(2*n-1)!!; a[n_, n_] = 1; Table[a[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 16 2012, after Johannes W. Meijer *)
-
row(n)=Vecrev(prod(i=1,n,'x-2*i+1)) \\ Charles R Greathouse IV, Feb 09 2017
A024196
a(n) = 2nd elementary symmetric function of the first n+1 odd positive integers.
Original entry on oeis.org
3, 23, 86, 230, 505, 973, 1708, 2796, 4335, 6435, 9218, 12818, 17381, 23065, 30040, 38488, 48603, 60591, 74670, 91070, 110033, 131813, 156676, 184900, 216775, 252603, 292698, 337386, 387005, 441905, 502448, 569008, 641971, 721735, 808710, 903318
Offset: 1
a(8) = 8*80+7*79+6*78+5*77+4*76+3*75+2*74+1*73 = 2796. - _Bruno Berselli_, Mar 13 2012
- Bruno Berselli, Table of n, a(n) for n = 1..1000
- Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Equals third right hand column of
A028338 triangle.
Equals third left hand column of
A109692 triangle.
Equals third right hand column of
A161198 triangle divided by 2^m.
(End)
-
List([1..36], n -> n*(n+1)*(3*n^2+5*n+1)/6); # Muniru A Asiru, Feb 13 2018
-
seq(n*(n+1)*(3*n^2+5*n+1)/6,n=1..25); # Muniru A Asiru, Feb 13 2018
-
f[k_] := 2 k - 1; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 50}] (* A024196 *)
(* Clark Kimberling, Dec 31 2011 *)
Table[(n(n+1)(3n^2+5n+1))/6,{n,50}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{3,23,86,230,505},50] (* Harvey P. Dale, Jul 08 2019 *)
A293318
a(n) = (2*n)! * [x^(2*n)] (-log(sqrt(1 - 2*x)))^n/(sqrt(1 - 2*x)*n!).
Original entry on oeis.org
1, 4, 86, 3480, 208054, 16486680, 1628301884, 192666441968, 26569595376038, 4184718381424152, 741138328282003860, 145795774074768177360, 31540994233548116475196, 7442380580681963411363440, 1902155375416975061879918520, 523496081998297020687019596000
Offset: 0
-
Table[(2 n)! SeriesCoefficient[(-Log[Sqrt[1 - 2 x]])^n/(Sqrt[1 - 2 x] n!), {x, 0, 2 n}], {n, 0, 15}]
A024198
4th elementary symmetric function of the first n+3 odd positive integers.
Original entry on oeis.org
105, 1689, 12139, 57379, 208054, 626934, 1646778, 3889578, 8439783, 17085783, 32645613, 59394517, 103613692, 174281212, 283927812, 449681892, 694529781, 1048818981, 1552033791, 2254874391, 3221672146, 4533175570, 6289743070
Offset: 1
- Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
- Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Equals fifth right hand column of
A028338 triangle.
Equals fifth left hand column of
A109692 triangle.
Equals fifth right hand column of
A161198 triangle divided by 2^m.
(End)
-
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{105,1689,12139,57379,208054,626934,1646778,3889578,8439783},30] (* Harvey P. Dale, May 28 2018 *)
-
Vec(-x*(x^4+112*x^3+718*x^2+744*x+105)/(x-1)^9 + O(x^100)) \\ Colin Barker, Aug 15 2014
A024197
a(n) = 3rd elementary symmetric function of the first n+2 odd positive integers.
Original entry on oeis.org
15, 176, 950, 3480, 10045, 24640, 53676, 106800, 197835, 345840, 576290, 922376, 1426425, 2141440, 3132760, 4479840, 6278151, 8641200, 11702670, 15618680, 20570165, 26765376, 34442500, 43872400, 55361475, 69254640, 85938426, 105844200, 129451505
Offset: 1
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
- Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Equals fourth right hand column of
A028338 triangle.
Equals fourth left hand column of
A109692 triangle.
Equals fourth right hand column of
A161198 triangle divided by 2^m.
(End)
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