A161198 -id:A161198 - cp's OEIS frontend

A024196 a(n) = 2nd elementary symmetric function of the first n+1 odd positive integers.

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

Clark Kimberling

			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).

From Johannes W. Meijer, Jun 08 2009: (Start)
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)
Cf. A016061.

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 *)

a(n) = n*(n+1)*(3*n^2+5*n+1)/6.
From Bruno Berselli, Mar 13 2012: (Start)
G.f.: x*(3 + 8*x + x^2)/(1 - x)^5.
a(n) = Sum_{i=1..n} (n+1-i)*((n+1)^2-i).
a(n) = n*A016061(n) - Sum_{i=0..n-1} A016061(i). (End)
a(n) - a(n-1) = A099721(n). Partial sums of A099721.- Philippe Deléham, May 07 2012
a(n) = Sum_{i=1..n} ((2*i-1)*Sum_{j=i..n} (2*j+1)) = 1*(3+5+...2*n+1) + 3*(5+7+...+2*n+1) + ... + (2*n-1)*(2*n+1). - J. M. Bergot, Apr 21 2017
a(n) = A028338(n+1, n-1), n >= 1, (third diagonal). See the crossref. below. Wolfdieter Lang, Jul 21 2017
a(n) = (A000583(n+1) - A000447(n+1))/2. - J. M. Bergot, Feb 13 2018

A074599 Numerator of 2 * H(n,2,1), a generalized harmonic number. See A075135. Also 2 * A350669.

Original entry on oeis.org

2, 8, 46, 352, 1126, 13016, 176138, 182144, 3186538, 62075752, 63461422, 1488711776, 7577414602, 23104065256, 680057071574, 21372905414144, 21646396991594, 21904260478904, 819482859775298, 828045249930848

Offset: 1

Robert G. Wilson v, Aug 27 2002

2*(1 + 1/3 + ... + 1/(2*n-1))/Pi = a(n)/(A350670(n)*Pi) is the equivalent resistance between the points (0,0) and (n,n) on a 2-dimension infinite square grid of unit resistors. - Jianing Song, Apr 28 2025

MathPages, The Algebra of an Infinite Grid of Resistors
Physics Stack Exchange, On this infinite grid of resistors, what's the equivalent resistance?

Cf. A350669. The denominators are in A350670.
Cf. A025547, A025550, A075135.
Not always equal to the second left hand column of A161198 triangle divided by A025549. - Johannes W. Meijer, Jun 08 2009

Table[ Numerator[ Sum[1/i, {i, 1/2, n}]], {n, 1, 20}]

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

A161199 Numerators in expansion of (1-x)^(-5/2).

Original entry on oeis.org

1, 5, 35, 105, 1155, 3003, 15015, 36465, 692835, 1616615, 7436429, 16900975, 152108775, 339319575, 1502700975, 3305942145, 115707975075, 251835004575, 1091285019825, 2354878200675, 20251952525805, 43397041126725, 185423721177825, 395033145117975

Offset: 0

Johannes W. Meijer, Jun 08 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A161198 (triangle for (1-x)^((-1-2*n)/2) for all values of n).
Cf. A046161 (denominators for (1-x)^(-5/2)).
Numerators of [x^n]( (1-x)^(p/2) ): A161202 (p=5), A161200 (p=3), A002596 (p=1), A001790 (p=-1), A001803 (p=-3), this sequence (p=-5), A161201 (p=-7).

A161199:= func< n | Numerator( Binomial(n+3,3)*Catalan(n+2)/2^(2*n+1) ) >;
[A161199(n): n in [0..30]]; // G. C. Greubel, Sep 24 2024

Numerator[CoefficientList[Series[(1-x)^(-5/2),{x,0,30}],x]] (* or *) Numerator[Table[(4n^2+8n+3)/3 Binomial[2n,n]/4^n,{n,0,30}]] (* Harvey P. Dale, Oct 15 2011 *)

def A161199(n): return numerator((-1)^n*binomial(-5/2,n))
[A161199(n) for n in range(31)] # G. C. Greubel, Sep 24 2024

a(n) = numerator(((3 + 8*n + 4*n^2)/3)*binomial(2*n,n)/(4^n)).

a(n) = denominator((3/2)*Integral_{x=0..1} x^n*sqrt(1-x) dx), where the integral is sqrt(Pi)*n!/Gamma(n+5/2) = n!/( (n+3/2)*(n+1/2)*(n-1/2)*...*(1/2)). - Groux Roland, Feb 23 2011

A161202 Numerators in expansion of (1-x)^(5/2).

Original entry on oeis.org

1, -5, 15, -5, -5, -3, -5, -5, -45, -55, -143, -195, -1105, -1615, -4845, -7429, -185725, -294975, -950475, -1550775, -10235115, -17058525, -57378675, -97294275, -1329688425, -2287064091, -7916760315, -13781027215

Offset: 0

Johannes W. Meijer, Jun 08 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A046161 (denominators).
Cf. A161198 (triangle of coefficients of (1-x)^((-1-2*n)/2)).
Numerators of [x^n]( (1-x)^(p/2) ): this sequence (p=5), A161200 (p=3), A002596 (p=1), A001790 (p=-1), A001803 (p=-3), A161199 (p=-5), A161201 (p=-7).

A161202:= func< n | -Numerator(15*(n+1)*Catalan(n)/(4^n*(2*n-1)*(2*n-3)*(2*n-5))) >;
[A161202(n): n in [0..30]]; // G. C. Greubel, Sep 24 2024

Numerator[CoefficientList[Series[(1-x)^(5/2),{x,0,30}],x]] (* Harvey P. Dale, Aug 22 2011 *)
Table[(-1)^n*Numerator[Binomial[5/2, n]], {n,0,30}] (* G. C. Greubel, Sep 24 2024 *)

def A161202(n): return (-1)^n*numerator(binomial(5/2,n))
[A161202(n) for n in range(31)] # G. C. Greubel, Sep 24 2024

a(n) = numerator( (15/(15-46*n+36*n^2-8*n^3))*binomial(2*n,n)/(4^n) ).

a(n) = (-1)^n*numerator( binomial(5/2, n) ). - G. C. Greubel, Sep 24 2024

A161201 Numerators in expansion of (1-x)^(-7/2).

Original entry on oeis.org

1, 7, 63, 231, 3003, 9009, 51051, 138567, 2909907, 7436429, 37182145, 91265265, 882230895, 2103781365, 9917826435, 23141595015, 856239015555, 1964313035685, 8948537162565, 20251952525805, 182267572732245

Offset: 0

Johannes W. Meijer, Jun 08 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A046161 (denominators).
Cf. A161198 (triangle of coefficients of (1-x)^((-1-2*n)/2)).
Numerators of [x^n]( (1-x)^(p/2) ): A161202 (p=5), A161200 (p=3), A002596 (p=1), A001790 (p=-1), A001803 (p=-3), A161199 (p=-5), this sequence (p=-7).

A161201:= func< n | Numerator((n+1)*(2*n+1)*(2*n+3)*(2*n+5)*Catalan(n)/(15*4^n)) >;
[A161201(n): n in [0..30]]; // G. C. Greubel, Sep 24 2024

CoefficientList[Series[(1-x)^(-7/2),{x,0,20}],x]//Numerator (* Harvey P. Dale, Jan 14 2020 *)
Table[(-1)^n*Numerator[Binomial[-7/2, n]], {n, 0, 30}] (* G. C. Greubel, Sep 24 2024 *)

def A161201(n): return (-1)^n*numerator(binomial(-7/2,n))
[A161201(n) for n in range(31)] # G. C. Greubel, Sep 24 2024

a(n) = numerator(((15+46*n+36*n^2+8*n^3)/15)*binomial(2*n,n)/(4^n)).

a(n) = (-1)^n*numerator( binomial(-7/2, n) ). - G. C. Greubel, Sep 24 2024

A028340 Coefficient of x^3 in expansion of (x+1)(x+3)...(x+2n-1).

Original entry on oeis.org

1, 16, 230, 3480, 57379, 1038016, 20570444, 444647600, 10431670821, 264300628944, 7198061846898, 209814739262856, 6520139954328519, 215245451727154944, 7524314127912551832, 277705505168550027360, 10792700030471840300745, 440604294676004639627280

Offset: 3

Bill Gosper

nonn

Equals fourth left hand column of A161198 triangle divided by 8. - Johannes W. Meijer, Jun 08 2009

G. C. Greubel, Table of n, a(n) for n = 3..400

Cf. A028338, A161198.

Table[Coefficient[Product[x + 2*k - 1, {k, 1, n}], x, 3], {n,3,50}] (* G. C. Greubel, Nov 24 2016 *)

a(n) = polcoeff(prod(k=1, n, x+2*k-1), 3); \\ Michel Marcus, Nov 12 2014

a(n) = Sum_{i=k+1..n} (-1)^(k+1-i)*2^(n-1)*binomial(i-1, k)*s1(n, i) with k = 3, where s1(n, i) are unsigned Stirling numbers of the first kind. - Victor Adamchik (adamchik(AT)ux10.sp.cs.cmu.edu), Jan 23 2001

E.g.f.: -(log(1-2*x))^3/( 48*sqrt(1-2*x) ). - Vladeta Jovovic, Feb 19 2003

More terms from Michel Marcus, Nov 12 2014

A028339 Coefficient of x^2 in expansion of (x+1)(x+3)...(x+2n-1).

Original entry on oeis.org

1, 9, 86, 950, 12139, 177331, 2924172, 53809164, 1094071221, 24372200061, 590546123298, 15467069396610, 435512515705695, 13121113142970855, 421214220916438680, 14354510691610713240, 517596339235489288425, 19688993487602867898225, 787995759739909824183150

Offset: 2

Bill Gosper

nonn

Equals third left hand column of A161198 triangle divided by 4. - Johannes W. Meijer, Jun 08 2009

			G.f. = x^2 + 9*x^3 + 86*x^4 + 950*x^5 + 12139*x^6 + 177331*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 2..400

Cf. A028338, A161198.

Table[Coefficient[Product[x + 2*k - 1, {k, 1, n}], x, 2], {n,2,50}] (* G. C. Greubel, Nov 24 2016 *)

a(n) = polcoeff(prod(k=1, n, x+2*k-1), 2); \\ Michel Marcus, Nov 12 2014

a(n) = Sum_{i=k+1,..,n}[ (-1)^(k+1-i) 2^(n-1) binomial(i-1, k) s1(n, i) ] with k = 2, where s1(n, i) are unsigned Stirling numbers of the first kind. - Victor Adamchik (adamchik(AT)ux10.sp.cs.cmu.edu), Jan 23 2001
E.g.f.: (log(1-2*x))^2/(8*sqrt(1-2*x)). - Vladeta Jovovic, Feb 19 2003
a(n) ~ n! * log(n)^2 * 2^(n-3) / sqrt(Pi*n) * (1 + (2*gamma + 4*log(2))/log(n)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 18 2017

More terms from Michel Marcus, Nov 12 2014

A028341 Coefficient of x^4 in expansion of (x+1)(x+3)...(x+2n-1).

Original entry on oeis.org

1, 25, 505, 10045, 208054, 4574934, 107494190, 2702025590, 72578259391, 2078757113719, 63324503917311, 2046225352864875, 69953125893139644, 2523698606200763196, 95853765344939263692, 3824294822931302783964, 159940198124792648875341, 6998152417792503243516261

Offset: 4

Bill Gosper

nonn

Equals fifth left hand column of A161198 triangle divided by 16. - Johannes W. Meijer, Jun 08 2009

			G.f. = x^4 + 25*x^5 + 505*x^6 + 10045*x^7 + 208054*x^8 + 4574934*x^9 + ...

Robert Israel, Table of n, a(n) for n = 4..369

Cf. A028338, A004041, A028339, A028340, A161198.

N:= 50: # to get a(4) to a(N)
P[0]:= 1;
for n from 1 to N do
  P[n]:= rem(P[n-1]*(x + 2*n-1), x^5,x)
od:
seq(coeff(P[n],x,4),n=4..N); # Robert Israel, Nov 13 2014

Table[Coefficient[Product[x + 2*k - 1, {k, 1, n}], x, 4], {n,4,50}] (* G. C. Greubel, Nov 24 2016 *)

a(n) = polcoeff(prod(k=1, n, x+2*k-1), 4); \\ Michel Marcus, Nov 12 2014

a(n) = Sum_{i=k+1,..,n} (-1)^(k+1-i)*2^(n-1)*binomial(i-1, k)*s1(n, i) with k = 4, where s1(n, i) are unsigned Stirling numbers of the first kind. - Victor Adamchik (adamchik(AT)ux10.sp.cs.cmu.edu), Jan 23 2001

E.g.f.: (log(1-2*x))^4/( 384*sqrt(1-2*x) ). - Vladeta Jovovic, Feb 19 2003

More terms from Michel Marcus, Nov 12 2014

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

Clark Kimberling

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.

From Johannes W. Meijer, Jun 08 2009: (Start)
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

a(n) = n*(n+1)*(n+2)*(n+3)*(15*n^4+150*n^3+515*n^2+672*n+223)/360.
G.f.: -x*(x^4+112*x^3+718*x^2+744*x+105) / (x-1)^9. - Colin Barker, Aug 15 2014
a(n) = A000332(n+3) * (15*n^4+150*n^3+515*n^2+672*n+223)/15 . - R. J. Mathar, Oct 01 2016
a(n) = A(n+4, n-1), n >= 1 (fifth diagonal). See a crossref. below. - Wolfdieter Lang, Jul 21 2017
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9). - Wesley Ivan Hurt, Jul 09 2025

cp's OEIS Frontend

A024196 a(n) = 2nd elementary symmetric function of the first n+1 odd positive integers.

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A074599 Numerator of 2 * H(n,2,1), a generalized harmonic number. See A075135. Also 2 * A350669.

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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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A161199 Numerators in expansion of (1-x)^(-5/2).

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A161202 Numerators in expansion of (1-x)^(5/2).

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A161201 Numerators in expansion of (1-x)^(-7/2).

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A028340 Coefficient of x^3 in expansion of (x+1)*(x+3)*...*(x+2*n-1).

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A028339 Coefficient of x^2 in expansion of (x+1)*(x+3)*...*(x+2*n-1).

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

A028340 Coefficient of x^3 in expansion of (x+1)(x+3)...(x+2n-1).

A028339 Coefficient of x^2 in expansion of (x+1)(x+3)...(x+2n-1).

A028341 Coefficient of x^4 in expansion of (x+1)(x+3)...(x+2n-1).