A028338 -id:A028338 - cp's OEIS frontend

A225470 Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

1, 2, 1, 10, 7, 1, 80, 66, 15, 1, 880, 806, 231, 26, 1, 12320, 12164, 4040, 595, 40, 1, 209440, 219108, 80844, 14155, 1275, 57, 1, 4188800, 4591600, 1835988, 363944, 39655, 2415, 77, 1, 96342400, 109795600, 46819324, 10206700, 1276009, 95200, 4186, 100, 1

Offset: 0

Peter Luschny, May 16 2013

The Stirling-Frobenius subset numbers S_{m}(n,k), for m >= 1 fixed, regarded as an infinite lower triangular matrix, can be inverted by Sum_{k} S_{m}(n,k)*s_{m}(k,j)*(-1)^(n-k) = [j = n]. The inverse numbers s_{m}(k,j), which are unsigned, are the Stirling-Frobenius cycle numbers. For m = 1 this gives the classical Stirling cycle numbers A132393. The Stirling-Frobenius subset numbers are defined in A225468.

Triangle T(n,k), read by rows, given by (2, 3, 5, 6, 8, 9, 11, 12, 14, 15, ... (A007494)) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 14 2015

			Triangle starts:
  [n\k][    0,      1,     2,     3,    4,  5,  6]
  [0]       1,
  [1]       2,      1,
  [2]      10,      7,     1,
  [3]      80,     66,    15,     1,
  [4]     880,    806,   231,    26,    1,
  [5]   12320,  12164,  4040,   595,   40,  1,
  [6]  209440, 219108, 80844, 14155, 1275, 57,  1.
  ...
From _Wolfdieter Lang_, Aug 11 2017: (Start)
Recurrence (see Maple program): T(4, 2) = T(3, 1) + (3*4 - 1)*T(3, 2) = 66 + 11*15 = 231.
Boas-Buck type recurrence for column k = 2 and n = 4: T(4, 2) = (4!/2)*(3*(2 + 6*(5/12))*T(2, 2)/2! + 1*(2 + 6*(1/2))*T(3,2)/3!) = (4!/2)*(3*9/4 + 5*15/3!) = 231. (End)

Peter Bala, A 3 parameter family of generalized Stirling numbers
Peter Luschny, Generalized Eulerian polynomials
Peter Luschny, The Stirling-Frobenius numbers

Cf. A225468; A132393 (m=1), A028338 (m=2), A225471(m=4).
Column k=0..4 give A008544, A024395(n-1), A286722(n-2), A383221, A383222.
T(n, n) ~ A000012; T(n, n-1) ~ A005449; T(n, n-2) ~ A024391; T(n, n-3) ~ A024392.
row sums ~ A032031; alternating row sums ~ A007559.
Cf. A132393.

SF_C := proc(n, k, m) option remember;
if n = 0 and k = 0 then return(1) fi;
if k > n or  k < 0 then return(0) fi;
SF_C(n-1, k-1, m) + (m*n-1)*SF_C(n-1, k, m) end:
seq(print(seq(SF_C(n, k, 3), k = 0..n)), n = 0..8);

SFC[0, 0, ] = 1; SFC[n, k_, ] /; (k > n || k < 0) = 0; SFC[n, k_, m_] := SFC[n, k, m] = SFC[n-1, k-1, m] + (m*n-1)*SFC[n-1, k, m]; Table[SFC[n, k, 3], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 26 2013, after Maple *)

For a recurrence see the Maple program.
From Wolfdieter Lang, May 18 2017: (Start)
This is the Sheffer triangle (1/(1 - 3*x)^{-2/3}, -(1/3)*log(1-3*x)). See the P. Bala link where this is called exponential Riordan array, and the signed version is  denoted by s_{(3,0,2)}.
E.g.f. of row polynomials in the variable x (i.e., of the triangle): (1 - 3*z)^{-(2+x)/3}.
E.g.f. of column k: (1-3*x)^(-2/3)*(-(1/3)*log(1-3*x))^k/k!, k >= 0.
Recurrence for row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k: R(n, x) = (x+2)*R(n-1,x+3), with R(0, x) = 1.
R(n, x) = risefac(3,2;x,n) := Product_{j=0..(n-1)} (x + (2 + 3*j)). (See the P. Bala link, eq. (16) for the signed s_{3,0,2} row polynomials.)
T(n, k) = Sum_{j=0..(n-m)} binomial(n-j, k)* S1p(n, n-j)*2^(n-k-j)*3^j, with S1p(n, m) = A132393(n, m). (End)
Boas-Buck type recurrence for column sequence k: T(n, k) = (n!/(n - k)) * Sum_{p=k..n-1} 3^(n-1-p)*(2 + 3*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), beginning {1/2, 5/12, 3/8, 251/720, ...}. See a comment and references in A286718. - Wolfdieter Lang, Aug 11 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

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

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

A225471 Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 3, 1, 21, 10, 1, 231, 131, 21, 1, 3465, 2196, 446, 36, 1, 65835, 45189, 10670, 1130, 55, 1, 1514205, 1105182, 290599, 36660, 2395, 78, 1, 40883535, 31354119, 8951355, 1280419, 101325, 4501, 105, 1, 1267389585, 1012861224, 308846124, 48644344, 4421494, 240856, 7756, 136, 1

Offset: 0

Peter Luschny, May 17 2013

The Stirling-Frobenius cycle numbers are defined in A225470.

Triangle T(n,k), read by rows, given by (3, 4, 7, 8, 11, 12, 15, 16, 19, 20, ... (A014601)) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 14 2015

			[n\k][    0,       1,      2,     3,    4,  5,  6 ]
[0]       1,
[1]       3,       1,
[2]      21,      10,      1,
[3]     231,     131,     21,     1,
[4]    3465,    2196,    446,    36,    1,
[5]   65835,   45189,  10670,  1130,   55,  1,
[6] 1514205, 1105182, 290599, 36660, 2395, 78,  1.
...
From _Wolfdieter Lang_, Aug 11 2017: (Start)
Recurrence: T(4, 2) = T(3, 1) + (4*4 - 1)*T(3, 2) = 131 +15*21 = 446.
Boas-Buck recurrence for column k=2 and n=4: T(4, 2) = (4!/2)*(4*(3+8*(5/12)) *T(2, 2)/2! + 1*(3 + 8*(1/2))*T(3,2)/3!) = (4!/2)*(4*(19/3)/2  + 7*21/3!) =  446.
(End)

P. Bala, A 3 parameter family of generalized Stirling numbers.
Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

Columns k=0..3 give A008545, A286723(n-1), A383702, A383703.

Cf. A132393 (m=1), A028338 (m=2), A225470 (m=3).

T[0, 0] = 1; T[n_, k_] := Sum[Binomial[n - j, k]*Abs[StirlingS1[n, n - j]]* 3^(n - k - j)*4^j, {j, 0, n - k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2018, after Wolfdieter Lang *)

@CachedFunction
def SF_C(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return SF_C(n-1, k-1, m) + (m*n-1)*SF_C(n-1, k, m)
for n in (0..8): [SF_C(n, k, 4) for k in (0..n)]

For a recurrence see the Sage program.
T(n, 0) ~ A008545; T(n, n) ~ A000012; T(n, n-1) = A014105.
Row sums ~ A047053; alternating row sums ~ A001813.
From Wolfdieter Lang, May 29 2017: (Start)
This is the Sheffer triangle (1/(1 - 4*x)^{-3/4}, -(1/4)*log(1-4*x)). See the P. Bala link where this is called exponential Riordan array, and the signed version is denoted by s_{(4,0,3)}.
E.g.f. of row polynomials in the variable x (i.e., of the triangle): (1 - 4*z)^{-(3+x)/4}.
E.g.f. of column k: (1-4*x)^(-3/4)*(-(1/4)*log(1-4*x))^k/k!, k >= 0.
Recurrence for row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k: R(n, x) = (x+3)*R(n-1,x+4), with R(0, x) = 1.
R(n, x) = risefac(4,3;x,n) := Product_{j=0..(n-1)} (x + (3 + 4*j)). (See the P. Bala link, eq. (16) for the signed s_{4,0,3} row polynomials.)
T(n, k) = Sum_{j=0..(n-m)} binomial(n-j, k)* S1p(n, n-j)*3^(n-k-j)*4^j, with S1p(n, m) = A132393(n, m).
T(n, k) = sigma[4,3]^{(n)}_{n-k}, with the elementary symmetric functions sigma[4,3]^{(n)}_m of degree m in the n numbers 3, 7, 11, ..., 3+4*(n-1), with sigma[4,3]^{(n)}_0 := 1. (End)
Boas-Buck type recurrence for column sequence k: T(n, k) = (n!/(n - k)) * Sum_{p=k..n-1} 4^(n-1-p)*(3 + 8*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), beginning with {1/2, 5/12, 3/8, 251/720, ...}. See a comment and references in A286718. - Wolfdieter Lang, Aug 11 2017

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

A286724 Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].

Original entry on oeis.org

1, 2, 1, 8, 8, 1, 48, 72, 18, 1, 384, 768, 288, 32, 1, 3840, 9600, 4800, 800, 50, 1, 46080, 138240, 86400, 19200, 1800, 72, 1, 645120, 2257920, 1693440, 470400, 58800, 3528, 98, 1, 10321920, 41287680, 36126720, 12042240, 1881600, 150528, 6272, 128, 1, 185794560, 836075520, 836075520, 325140480, 60963840, 6096384, 338688, 10368, 162, 1, 3715891200, 18579456000, 20901888000, 9289728000, 2032128000, 243855360, 16934400, 691200, 16200, 200, 1

Offset: 0

Wolfdieter Lang, Jun 16 2017

These generalized unsigned Lah numbers are the instance L[2,1] of the Sheffer triangles called L[d,a], with integers d >= 1 and integers 0 <= a < d with gcd(d,a) = 1. The standard unsigned Lah numbers are L[1,0] = A271703.
The Sheffer structure of L[d,a] is ((1 - d*t)^(-2*a/d), t/(1 - d*t)). This follows from the defining property
  risefac[d,a](x, n) = Sum_{m=0..n} L[d,a](n, m)*fallfac[d,a](x, m), where risefac[d,a](x, n):= Product_{0..n-1} (x  + (a+d*j)) for n >= 1 and risefac[d,a](x, 0) := 1, and fallfac[d,a](x, n):= Product_{0..n-1} (x - (a+d*j)) = for n >= 1 and fallfac[d,a](x, 0) := 1. Such rising and falling factorials arise in the generalization of Stirling numbers of both kinds S2[d,a] and S1[d,a]. See the Peter Bala link under A143395 for these falling factorials called there [t;a,b,c]_n with t=x, a=d, b=0, c=a.
In matrix notation: L[d,a] = S1phat[d,a].S2hat[d,a] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations with Sheffer structures S1phat[d,a] = ((1 - d*t)^(-a/d), -(1/d)*(log(1 - d*t))) and S2hat[d,a] = (exp(a*t), (1/d)*(exp(d*t) - 1). See, e.g., S1phat[2,1] = A028338 and S2hat[2,1] = A039755.
The a- and z-sequences for these Sheffer matrices have e.g.f.s 1 + d*t and ((1 + d*t)/t)*(1 - (1 + d*t)^(-2*a/d)), respectively. See a W. Lang link under A006232 for these types of sequences.
E.g.f. of row polynomials R[d,a](n, x) := Sum_{m=0..n} L[d,a](n, m)*x^m
  (1 - d*x)^(-2*a/d)*exp(t*x/(1 - d*x)) (this is the e.g.f. for the triangle).
E.g.f. of column m: (1 - d*t)^(-2*a/d)*(t/(1 - d*t))^m/m, m >= 0.
Meixner type identity for (monic) row polynomials: (D_x/(1 + d*D_x)) * R[d,a](n, x) = n*R[d,a](n-1, x), n >= 1, with R[d,a](0, x) = 1. The series in the differentiations D_x = d/dx terminates.
General Sheffer recurrence for row polynomials (see the Roman reference, p. 50, Corollary 3.7.2, rewritten for the present Sheffer notation):
  R[d,a](n, x) = [(2*a+x)*1 + 2*d*(a + x)*D_x + d^2*x*(D_x)^2]*R[d,a](n-1, x), n >= 1, with R[d,a](0, x) = 1.
The inverse matrix L^(-1)[d,a] is Sheffer (g[d,a](-t), -f[d,a](-t)) with L[d,a] Sheffer (g[d,a](t), f[d,a](t)) from above. This means (see the column e.g.f. of Sheffer matrices) that L^(-1)[d,a](n, m) = (-1)^(n-m)*L[d,a](n, m). Therefore, the recurrence relations can easily be rewritten for L^(-1)[d,a] by replacing a -> -a and d -> -d.
fallfac[d,a](x, n) = Sum_{m=0..n} L^(-1)[d,a](n, m)*risefac[d,a](x, m), n >= 0.
From Wolfdieter Lang, Aug 12 2017: (Start)
The Sheffer row polynomials R[d,a](n, x) belong to the Boas-Buck class and satisfy therefore the Boas-Buck identity (see the reference, and we use the notation of Rainville, Theorem 50, p. 141, adapted to an exponential generating function) (E_x - n*1)*R[d,a](n, x) = - n*(2*a*1 + d*E_x) * Sum_{k=0..n-1} d^k*R(d,a;n-1-k,x)/(n-1-k)!, with E_x = x*d/dx (Euler operator).
This implies a recurrence for the sequence of column m: L[d,a](n, m) = (n!*(2*a + d*m)/(n-m))*Sum_{p=0..n-1-m} d^p*L[d,a](n-1-p, m)/(n-1-p)!, for n > m>=0, and input L[d,a](m, m) = 1. For the present [d,a] = [2,1] instance see the formula and example sections. (End)
From Wolfdieter Lang, Sep 14 2017: (Start)
The diagonal sequences are 2^D*D!*(binomial(m+D, m))^2, m >= 0, for D >= 0 (main diagonal D = 0). From the o.g.f.s obtained via Lagrange's theorem. See the second W. Lang link below for the general Sheffer case.
The o.g.f. of the diagonal D sequence is 2^D*D!*Sum_{m=0..D} A008459(D, m)*x^m /(1- x)^(2*D + 1),  D >= 0. (End)
It appears that this is also the matrix square of unsigned triangle of coefficients of Laguerre polynomials n!*L_n(x), abs(A021009(n, k)). - Ali Pourzand, Mar 10 2025 [This observation is correct. - Peter Luschny, Mar 10 2025]

			The triangle T(n, m) begins:
  n\m        0         1         2         3        4       5      6     7   8 9
  0:         1
  1:         2         1
  2:         8         8         1
  3:        48        72        18         1
  4:       384       768       288        32        1
  5:      3840      9600      4800       800       50       1
  6:     46080    138240     86400     19200     1800      72      1
  7:    645120   2257920   1693440    470400    58800    3528     98     1
  8:  10321920  41287680  36126720  12042240  1881600  150528   6272   128   1
  9: 185794560 836075520 836075520 325140480 60963840 6096384 338688 10368 162 1
  ...
From _Wolfdieter Lang_, Aug 12 2017: (Start)
Recurrence for column elements with m >= 1, and input column m = 0: T(3, 2) = (3/2)*T(2, 1) + 2*3*T(2, 2) = (3/2)*8 + 6 = 18.
Four term recurrence: T(3, 2) = T(2, 1) + 2*5*T(2, 2) - 4*2^2*T(1, 2) = 8 + 10 + 0 = 18.
Meixner type identity, n=2: 2*R(1, x) = (D_x - 2*(D_x)^2)*R(2, x), 2*(2 + x) = (8 + 2*x) - 2*2.
Sheffer recurrence: R(2, x) = (2 + x)*(2 + x) + 4*(1 + x)*1 + 0 = 8 + 8*x + x^2.
Boas-Buck recurrence for column m = 2 and n = 4: T(4, 2) = (2*4!*3/2)*(1*T(3, 2)/3! + 2*T(2, 2)/2!) = 4!*3*(18/3! + 1) = 288. (End)
Diagonal sequence D = 1: o.g.f. 2*1!*(1 + 1*x)/(1- x)^3 generating
{2*(binomial(m+1, m))^2}_{m >= 0} = {2, 8, 18, 32, ...}. - _Wolfdieter Lang_, Sep 14 2017

Ralph P. Boas, jr. and R. Creighton Buck, Polynomial Expansions of analytic functions, Springer, 1958, pp. 17 - 21, (last sign in eq. (6.11) should be -).
Earl D. Rainville, Special Functions, The Macmillan Company, New York, 1960, ch. 8, sect. 76, 140 - 146.
Steven Roman, The Umbral Calculus, Academic press, Orlando, London, 1984, p. 50.

Peter Bala, The white diamond product of power series
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.
Wolfdieter Lang, On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.

Cf. A006232, A025167, A028338, A039755, A143395, A271703, A021009.
Column sequences (no leading zeros): A000165, A014479, A286725.
Diagonal sequences: A000012, 2*A000290(m+1), 8*A000537(n+1), 48*A001249, 384*A288876. - Wolfdieter Lang, Sep 14 2017
Row sums are A025167. - Michael Somos, Sep 27 2017

T := (n, k) -> ifelse(n < k, 0, ifelse(k = 0, n!*2^n, (n/k)*T(n-1, k-1) + 2*n*T(n-1, k))): seq(seq(T(n, k), k = 0..n), n = 0..10);  # Peter Luschny, Mar 10 2025

T[ n_, k_] := Coefficient[ Integrate[ Exp[-x^2 - y x] HermiteH[n, x]^2, {x, -Infinity, Infinity}] / (Sqrt[Pi] Exp[y^2 / 4]), y, 2 k]; (* Michael Somos, Sep 27 2017 *)

# Using the function A021009_triangle, displays as a matrix. Following the observation of Ali Pourzand.
print(A021009_triangle(9)^2)  # Peter Luschny, Mar 10 2025

T(n, m) = L[2,1](n, m) = Sum_{k=m..n} A028338(n, k)*A039755(k, m).
Three term recurrence for column elements with m >= 1: T(n, m) = (n/m)*T(n-1, m-1) + 2*n*T(n-1, m) with T(n, m) = 0 for n < m and the column m = 0 is T(n, 0) = (2*n)!! = n*2^n = A000165(n). (From the a- and z-sequences {1, 2, repeat(0)} and {2, repeat(0)}, respectively.)
Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(2*n-1)*T(n-1, m) - 4*(n-1)^2*T(n-2, m), n >= m >= 0, with T(0, 0) = 1, T(-1, m) = 0, T(n, -1) = 0 and T(n, m) = 0 if n < m.
E.g.f. of row polynomials R(n, x) = R[2,1](n, x) (i.e., e.g.f. of the triangle): (1/(1-2*t))*exp(x*t/(1-2*t)).
E.g.f. of column m sequences: (t^m/(1-2*t)^(m+1))/m!, m >= 0.
Meixner type identity: Sum_{k=0..n-1} (-1)^k*2^k*(D_x)^(k+1)*R(n, x) = n*R(n-1, x), n >= 1, with R(0, x) = 1 and D_x = d/dx.
Sheffer recurrence: R(n, x) = [(2 + x)*1 + 4*(1 + x)*D_x + 4*x*(D_x)^2]*R(n-1, x), n >= 1, and R(0, x) = 1.
Boas-Buck recurrence for column m (see a comment above): T(n, m) = (2*n!*(1 + m)/(n-1))*Sum_{p=0..n-1-m} 2^p*T(n-1-p, m)/(n-1-p)!, for n > m >= 0, and input T(m, m) = 1. - Wolfdieter Lang, Aug 12 2017
Explicit form (from the diagonal sequences with the o.g.f.s given as a comment above): T(n, m) = 2^(n-m)*(n-m)!*(binomial(n, n-m))^2 for n >= m >= 0. - Wolfdieter Lang, Sep 23 2017
Let R(n,x) denote the n-th row polynomial. Then x^n*R(n,x) = x^n o x^n, where o denotes the deformed Hadamard product of power series defined in Bala, Section 3.1. - Peter Bala, Jan 18 2018

A293318 a(n) = (2n)! [x^(2n)] (-log(sqrt(1 - 2x)))^n/(sqrt(1 - 2x)n!).

Original entry on oeis.org

1, 4, 86, 3480, 208054, 16486680, 1628301884, 192666441968, 26569595376038, 4184718381424152, 741138328282003860, 145795774074768177360, 31540994233548116475196, 7442380580681963411363440, 1902155375416975061879918520, 523496081998297020687019596000

Offset: 0

Ilya Gutkovskiy, Oct 06 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..300

Central terms of triangles A028338, A039757 (gives absolute values) and A109692.

Cf. A265846.

Table[(2 n)! SeriesCoefficient[(-Log[Sqrt[1 - 2 x]])^n/(Sqrt[1 - 2 x] n!), {x, 0, 2 n}], {n, 0, 15}]

a(n) ~ c * d^n * (n-1)!, where d = -16*LambertW(-1, -exp(-1/2)/2)^2 / (1 + 2*LambertW(-1, -exp(-1/2)/2)) = 19.643259858273023595... (see also A265846) and c = 1/(2*Pi*sqrt(1 + 1/LambertW(-1, -exp(-1/2)/2))) = 0.2425219128152359859... - Vaclav Kotesovec, Oct 18 2017, updated Mar 17 2024 and May 14 2025

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

A225470 Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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

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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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A225471 Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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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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A028341 Coefficient of x^4 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).

A293318 a(n) = (2n)! [x^(2n)] (-log(sqrt(1 - 2x)))^n/(sqrt(1 - 2x)n!).