A112007 -id:A112007 - cp's OEIS frontend

A290311 Triangle T(n, k) read by rows: row n gives the coefficients of the row polynomials of the (n+1)-th diagonal sequence of the Sheffer triangle A094816 (special Poisson-Charlier).

1, 1, 0, 1, 3, -1, 1, 17, -2, -1, 1, 80, 49, -27, 2, 1, 404, 733, -153, -49, 9, 1, 2359, 7860, 1622, -1606, 150, 9, 1, 16057, 80715, 58965, -17840, -3876, 1163, -50, 1, 125656, 858706, 1150722, 47365, -175756, 18239, 2359, -267, 1, 1112064, 9710898, 19571174, 7548463, -3175846, -491809, 194777, -9884, -413

Offset: 0

Wolfdieter Lang, Jul 28 2017

The o.g.f. of the (n+1)-th diagonal sequence of the Sheffer triangle (e^x, -(log(1-x))) (the product of two Sheffer triangles A007318*A132393 = Pascal*|Stirling1|) is P(n, x)/(1 - x)^{2*n+1}, for n >= 0., with the numerator polynomials P(n, x) = Sum_{k=0..n} T(n, k)*x^k.

O.g.f.'s for diagonal sequences of Sheffer matrices (lower triangular) can be computed via Lagrange's theorem. For the special case of Jabotinsky matrices (1, f(x)) this has been done by P. Bala (see the link under A112007), and the method can be generalized to Sheffer (g(x), f(x)), as shown in the W. Lang link given below.

			The triangle T(n, k) begins:
n\k 0       1       2        3       4        5       6      7     8    9 ...
0:  1
1:  1       0
2:  1       3      -1
3:  1      17      -2       -1
4:  1      80      49      -27       2
5:  1     404     733     -153     -49        9
6:  1    2359    7860     1622   -1606      150       9
7:  1   16057   80715    58965  -17840    -3876    1163    -50
8:  1  125656  858706  1150722   47365  -175756   18239   2359  -267
9:  1 1112064 9710898 19571174 7548463 -3175846 -491809 194777 -9884 -413
...
n = 2: the o.g.f. of the third diagonal of triangle A094816, [1, 8, 29, 75, 160, ...] = A290312 is  (1 + 3*x - x^2)/(1 - x)^5.

Wolfdieter Lang, On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.

Cf. A094816, A290312, A290313, A290314.

rows = 10; nmax = 30(*terms to find every gf*);
T = Table[(-1)^(n - k) Sum[Binomial[-j - 1, -n - 1] StirlingS1[j, k], {j, 0, n}], {n, 0, nmax}, {k, 0, nmax}];
row[n_] := FindGeneratingFunction[Diagonal[T, -n], x] // Numerator // CoefficientList[-#, x]&; row[0] = {1}; row[1] = {1, 0};
Table[row[n], {n, 0, rows-1}] // Flatten (* Jean-François Alcover, Jan 26 2019 *)

T(n, k) = [x^n] P(n, x) with the numerator polynomials (in rising powers) of the o.g.f. of the (n+1)-th diagonal sequence of the triangle A094816. See the comment above.

A112002 Seventh diagonal of triangle A008275 (Stirling1) and seventh column of |A008276|.

Original entry on oeis.org

720, 13068, 118124, 723680, 3416930, 13339535, 44990231, 135036473, 368411615, 928095740, 2185031420, 4853222764, 10246937272, 20692933630, 40171771630, 75289668850, 136717357942, 241276443496, 414908513800, 696829576300

Offset: 1

Wolfdieter Lang, Sep 12 2005

T. D. Noe, Table of n, a(n) for n = 1..1000

Sixth diagonal A053567; A130534.

[StirlingFirst(n+6, n): n in [1..20]]; // Vincenzo Librandi, Aug 09 2015

A112002 := proc(n) combinat[stirling1](n+6,n) ; end proc: # R. J. Mathar, Jun 08 2011

Table[StirlingS1[n+6, n], {n, 1, 20}] (* Jean-François Alcover, Mar 05 2014 *)

[stirling_number1(n,n-6) for n in range(7, 27)] # Zerinvary Lajos, May 16 2009

a(n)= Stirling1(n+6, n), n>=1, with Stirling1(n, k)= A008275(n, k).
E.g.f. with offset 6: exp(x)*sum(A112486(6, m)*(x^(6+m))/(6+m)!, m=0..6).
a(n)= (f(n+5, 6)/12!)*sum(A112486(6, m)*f(12, 6-m)*f(n-1, m), m=0..min(6, n-1)), with the falling factorials f(n, k):=n*(n-1)*...*(n-(k-1)). From the e.g.f.
a(n)=(binomial(n+6, 7)/r(8, 5))*sum(A112007(5, m)*r(n+7, 5-m)*f(n-1, m), m=0..5), with rising factorials r(n, k):=n*(n+1)*...*(n+(k-1)) and falling factorials f(n, m). From the g.f.
G.f.: x*(720+3708*x+4400*x^2+1452*x^3+114*x^4+x^5)/(1-x)^13. See row k=5 of triangles A112007 or A008517 for the coefficients.
Explicit formula: a(n) = n(n + 1)(n + 2)(n + 3)(n + 4)(n + 5)(n + 6)(63n^5 + 1575n^4 + 15435n^3 + 73801n^2 + 171150n + 152696)/2903040. - Vaclav Kotesovec, Jan 30 2010

A135338 Triangle read by rows: row n gives coefficients C(n,j) for a Sheffer sequence (binomial-type) with raising operator -x { 1 + W[ -exp(-2) * (2+D) ] } where W is the Lambert W multi-valued function.

Original entry on oeis.org

1, -1, 1, 1, -3, 1, -2, 7, -6, 1, 6, -20, 25, -10, 1, -24, 76, -105, 65, -15, 1, 120, -364, 511, -385, 140, -21, 1, -720, 2108, -2940, 2401, -1120, 266, -28, 1, 5040, -14328, 19720, -16632, 8841, -2772, 462, -36, 1, -40320, 111816, -151620, 129340, -73605, 27237, -6090, 750, -45, 1

Offset: 1

Tom Copeland, Feb 15 2008

The lowering (or delta) operator for these polynomials is L = -1 + exp{ 2 + W[ -exp(-2) * (2+D) ] } = Sum_{j >= 1} A074059(j) * D^j / j!.
The raising operator is R = -x { 1 + W[ -exp(-2) * (2+D) ] } = x { 1 + Sum_{j >= 1} (-1)^j * PW(j-1,-2) * D^j / j! }, where PW(j-1,x) are the polynomials of A042977.
W(x) here is W_-1 in the Monir reference and, about x = 0, W[ -exp(-2) * (2+x) ] = -[ 2 + Sum_{j >= 1} (-1)^j * PW(j-1,-2) * x^j / j! ].
From the relation between the delta and raising operators for associated binomial-type polynomials, A074059 = (1,1,2,7,34,...) and S = (1,-PW(0,-2),PW(1,-2),-PW(2,-2),...) = (1, -1, 0, -1, -2, -13, -74, -593, -5298, ...) form a list partition transform pair (see A133314); i.e., S and A074059 have reciprocal e.g.f.s and satisfy mutual recursion relations. Applying Faa di Bruno's formula to L gives other interesting integer relations between S and A074059.
The Bell transform of (-1)^n*factorial(n-1) if n>0, else 1. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle read by rows:
     1;
    -1,    1;
     1,   -3,     1;
    -2,    7,    -6,    1;
     6,  -20,    25,  -10,     1;
   -24,   76,  -105,   65,   -15,   1;
   120, -364,   511, -385,   140, -21,   1;
  -720, 2108, -2940, 2401, -1120, 266, -28, 1;
...
From _R. J. Mathar_, Mar 22 2013: (Start)
The matrix inverse starts:
     1;
     1,    1;
     2,    3,    1;
     7,   11,    6,   1;
    34,   55,   35,  10,   1;
   213,  349,  240,  85,  15,  1;
  1630, 2695, 1939, 770, 175, 21, 1;
  ... (End)

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
F. Chapeau-Blondeau and A. Monir, Numerical Evaluation of the Lambert W Function and Application to Generation of Generalized Gaussian Noise With Exponent 1/2, IEEE Trans. on Signal Processing, Vol. 50, No. 9, Sept. 2002, p. 2160-2164.

Cf. A134685, A135494.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n=0,1,(-1)^n*(n-1)!), 9); # Peter Luschny, Jan 27 2016

max = 10; s = Series[Exp[t*(2*x-(1+x)*Log[1+x])], {x, 0, max}, {t, 0, max}] // Normal; c[n_, j_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, j}]*n!; Table[c[n, j], {n, 1, max}, {j, 1, n}] // Flatten (* Jean-François Alcover, Apr 23 2014, after Peter Bala, duplicate of Copeland's e.g.f. *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[Function[n, If[n == 0, 1, (-1)^n (n-1)!]], rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 26 2018, after Peter Luschny *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: (-1)^n*factorial(n-1) if n>0 else 1, 10) # Peter Luschny, Jan 18 2016

The row polynomials P(n,t) = Sum_{j=1..n} C(n,j) * t^j satisfy exp[P(.,t) * x] = exp{ -t * [(1+x) * log(1+x) - 2*x] }, with P(0,t) = 1 and [ P(.,x) + P(.,y) ]^n = P(n,x+y). Here, as in the e.g.f., the umbral maneuver P(.,t)^n = P(n,t) is assumed. See Mathworld and Wikipedia on Sheffer sequences and umbral calculus for other general formulas, including expansion theorems.
From Peter Bala, Dec 09 2011: (Start)
E.g.f.: exp(t*(2*x-(1+x)*log(1+x))) = 1 + t*x + (t^2-t)*x^2/2! + (t^3-3*t^2+t)*x^3/3! + ... (Restatement of Copeland's e.g.f. above in umbral notation with P(.,t)^n = P(n,t).).
If a triangular array has an e.g.f. of the form exp(t*F(x)) with F(0) = 0, then the o.g.f.'s for the diagonals of the triangle are rational functions in t (see the Bala link). The rational functions are the coefficients in the compositional inverse (with respect to x) (x-t*F(x))^(-1). In this case (x-t*(2*x-(1+x)*log(1+x)))^(-1) = x/(1-t) - t/(1-t)^3*x^2/2! + (t+2*t^2)/(1-t)^5*x^3/3! - (2*t+6*t^2+7*t^3)/(1-t)^7*x^4/4! + ... . So, for example, the (unsigned) third subdiagonal has o.g.f. (2*t+6*t^2+7*t^3)/(1-t)^7 = 2*t + 20*t^2 + 105*t^3 + 385*t^4 + ... .
(End)

More terms from Jean-François Alcover, Apr 23 2014

A202017 Triangle of coefficients of the numerator polynomials of the rational o.g.f.'s of the diagonals of A059297.

Original entry on oeis.org

1, 2, 3, 9, 4, 52, 64, 5, 195, 855, 625, 6, 606, 6546, 15306, 7776, 7, 1701, 38486, 201866, 305571, 117649, 8, 4488, 194160, 1950320, 6244680, 6806472, 2097152, 9, 11367, 887949, 15597315, 90665595, 200503701, 168205743, 43046721

Offset: 1

Peter Bala, Dec 08 2011

If a triangular array has an e.g.f. of the form exp(t*F(x)) with F(0) = 0, then the o.g.f.'s for the diagonals of the triangle are rational functions in t [Drake, 1.10]. The rational functions are the coefficients in the compositional inverse (with respect to x) (x-t*F(x))^(-1).

Triangle A059297 has e.g.f. exp(t*x*exp(x)). The present triangle lists the coefficients of the numerator polynomials of the rational o.g.f.'s of the diagonals of A059297. Drake, Example 1.10.9, gives three combinatorial interpretations for these coefficients (but note the expansion at the bottom of p.68 is for (x-t*(-W(-x))^(-1), W(x) the Lambert W function, and not for (x-t*x*exp(x))^(-1) as stated there). Row reversal of A155163.

			Triangle begins
..n\k.|...1.....2......3.......4.......5.......6
= = = = = = = = = = = = = = = = = = = = = = = =
..1..|...2
..2..|...3.....9
..3..|...4....52.....64
..4..|...5...195....855.....625
..5..|...6...606...6546...15306....7776
..6..|...7..1701..38486..201866..305571..117649
...

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.

Cf. A059297, A155163 (row reverse).

T(n,k) = sum {j = 0..k} (-1)^(k-j)*C(2*n+1,k-j)*C(n+j,j)*j^n.

The compositional inverse (with respect to x) (x-t*x*exp(x))^-1 = x/(1-t) + 2*t/(1-t)^3*x^2/2! + (3*t+9*t^2)/(1-t)^5*x^3/3! + (4*t+52*t^2+64*t^3)/(1-t)^7*x^4/4! + .... The numerator polynomials begin 1, 2*t, (3*t+9*t^2), .... The initial 1 has been omitted from the array. Row sums appear to be A001813.

A288874 Row reversed version of triangle A201637 (second-order Eulerian triangle).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 8, 1, 0, 24, 58, 22, 1, 0, 120, 444, 328, 52, 1, 0, 720, 3708, 4400, 1452, 114, 1, 0, 5040, 33984, 58140, 32120, 5610, 240, 1, 0, 40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0, 362880, 3733920, 11026296, 12440064, 5765500, 1062500, 67260, 1004, 1, 0, 3628800, 44339040, 162186912, 238904904, 155357384, 44765000, 5326160, 218848, 2026, 1

Offset: 0

Wolfdieter Lang, Jul 20 2017

See A201637, and also A008517 (offset 1 for rows and columns).
The row polynomials of this triangle P(n, x) = Sum_{m=0..n} T(n, m)*x^m appear as numerator polynomials in the o.g.f.s for the diagonal sequences of triangle A132393 (|Stirling1| with offset 0 for rows and columns). See the comment and the P. Bala link there.
For similar triangles see also A112007 and A163936.

			The triangle T(n, m) begins:
n\m 0      1       2        3        4       5       6     7    8  9 ...
0:  1
1:  0      1
2:  0      2       1
3:  0      6       8        1
4:  0     24      58       22        1
5:  0    120     444      328       52       1
6:  0    720    3708     4400     1452     114       1
7:  0   5040   33984    58140    32120    5610     240     1
8:  0  40320  341136   785304   644020  195800   19950   494    1
9:  0 362880 3733920 11026296 12440064 5765500 1062500 67260 1004  1
...

Seiichi Manyama, Rows n = 0..139, flattened
Andrew Elvey Price, Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
Wikipedia, Eulerian numbers of the second kind

Cf. A201637, A008517, A112007, A163936.
Columns m = 0..5: A000007, A000142, A002538, A002539, A112008, A112485.
Diagonals d = 0..3: A000012, A005803, A004301, A006260.
T(2n,n) gives A290306.

T:= (n, k)-> combinat[eulerian2](n, n-k):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Jul 26 2017
# Using the e.g.f:
alias(W = LambertW): len := 10:
egf := (t - 1)*(1/(W(-exp(((t - 1)^2*x - 1)/t)/t) + 1) - 1):
ser := simplify(subs(W(-exp(-1/t)/t) = (-1/t), series(egf, x, len+1))):
seq(seq(n!*coeff(coeff(ser, x, n), t, k), k = 0..n), n = 0..len);  # Peter Luschny, Mar 13 2025

Table[Boole[n == 0] + Sum[(-1)^(n + k) * Binomial[2 n + 1, k] StirlingS1[2 n - m - k, n - m - k], {k, 0, n - m - 1}], {n, 0, 10}, {m, n, 0, -1}] // Flatten (* Michael De Vlieger, Jul 21 2017, after Jean-François Alcover at A201637 *)

T(n, m) = A201637(n, n-m), n >= m >= 0.
Recurrence: T(0, 0) = 1, T(n, -1) = 0, T(n, m) = 0 if n < m, (n-m+1)*T(n-1, m-1) + (n-1+m)*T(n-1, m), n >= 1, m = 0..n; from the A008517 recurrence.
T(0, 0) = 1, T(n, m) = Sum_{p = 0..m-1} (-1)^(n-p)*binomial(2*n+1, p)*A132393(n+m-p, m-p), n >= 1, m = 0..n; from a A008517 program.
T(n, k) = n! * [t^k][x^n] (t - 1)*(1/(W(-exp(((t - 1)^2*x - 1)/t)/t) + 1) - 1) where after expansion W(-exp(-1/t)/t) is substituted by (-1/t). [Inspired by a formula of Shamil Shakirov in A008517.] - Peter Luschny, Mar 13 2025

A288875 Triangle read by rows. The rows give the coefficients of the numerator polynomials for the o.g.f.s of the diagonal sequences of triangle A028338.

Original entry on oeis.org

1, 1, 1, 3, 8, 1, 15, 71, 33, 1, 105, 744, 718, 112, 1, 945, 9129, 14542, 5270, 353, 1, 10395, 129072, 300291, 191384, 33057, 1080, 1, 135135, 2071215, 6524739, 6338915, 2033885, 190125, 3265, 1, 2027025, 37237680, 150895836, 204889344, 103829590, 18990320, 1038780, 9824, 1

Offset: 0

Wolfdieter Lang, Jul 21 2017

The Sheffer triangle  A028338 of the type (1/sqrt(1-2*x), -(1/2)*log(1 - 2*x)) is called here |S1hat[2,1]|. The o.g.f. of the sequence of diagonal d, d >= 0 is D(d, t) = Sum_{m=0..d} A028338(d+m, m)*t^m. The e.g.f. of these o.g.f.s is taken as ED(y,t) := Sum_{d >= 0} D(d, t)*y^(d+1)/(d+1)!.
This e.g.f. is found to be ED(y,t) = 1 - sqrt(1 - 2*x(t;y)), where x = x(t;y) is the compositional inverse of y = y(t;x) = x*(1 - t*(-log(1-2*x)/(2*x))) = x + t*log(1-2*x)/2. The o.g.f.s are then D(d, t) = P(d, t)/(1 - t)^(2*d+1), with the row polynomials P(d, t) = Sum_{m=0..d} T(d, m)*t^m, d >= 0.
This computation was inspired by an article of P. Bala (see a link under, e.g., A112007) for Sheffer triangles of the Jabotinsky type (1, F(x)). There Sheffer is called exponential Riordan, and the diagonals are labeled by n = d+1, n >= 1.

			The triangle T(n, m) begins:
n\m      0        1         2         3         4        5       6    7  8 ...
0:       1
1:       1        1
2:       3        8         1
3:      15       71        33         1
4:     105      744       718       112         1
5:     945     9129     14542      5270       353        1
6:   10395   129072    300291    191384     33057     1080       1
7:  135135  2071215   6524739   6338915   2033885   190125    3265    1
8: 2027025 37237680 150895836 204889344 103829590 18990320 1038780 9824  1
...

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
Wolfdieter Lang, On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.

Cf. A028338, A112007, A214406.

De[d_, t_] := Sum[A028338[d+m, m] t^m, {m, 0, d}]; A028338[n_, k_] := SeriesCoefficient[Times @@ Table[x+i, {i, 1, 2n-1, 2}], {x, 0, k}]; P[n_, x_] := De[n, x] (1-x)^(2n+1); T[n_, m_] := Coefficient[P[n, x], x, m]; Table[T[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 24 2017 *)
T[n_,m_]:=Sum[(-1)^(i-n+m)*Binomial[2*n+1,n-m-i]*(1/(2^i*i!)*Sum[(-1)^(i-j)*Binomial[i,j]*(2*j+1)^(n+i),{j,0,i}]),{i,0,n-m}];Flatten[Table[T[n,m],{n,0,8},{m,0,n}]] (* Detlef Meya, Dec 18 2023, after Peter Bala from A214406 *)

T(n, m) = [x^m] P(n, x), with the numerator polynomial of the o.g.f. of the diagonal n (main diagonal n=0) D(n, x) = P(n, x)/(1-x)^(2*n+1). See a comment above.

T(n, m) = Sum_{i=0..n-m} ( (-1)^(i-n+m)*binomial(2*n+1,n-m-i)*(1/(2^i*i!))*Sum_{j=0..i} (-1)^(i-j)*binomial(i,j)*(2*j+1)^(n+i) ). - Detlef Meya, Dec 18 2023, after Peter Bala from A214406.

A290306 Number of permutations of the multiset {1,1,2,2,...,2n,2n} having exactly n ascents and no number smaller than k between the two occurrences of any number k.

Original entry on oeis.org

1, 2, 58, 4400, 644020, 155357384, 56041398784, 28299910066112, 19076135772884080, 16558710676700081120, 17997592513561138205728, 23948993629880321407298816, 38303802347672648465676584704, 72510806370598644118983905976320, 160368191672482402606757066578885120

Offset: 0

Alois P. Heinz, Jul 26 2017

nonn

			a(1) = 2: 1122, 1221.
a(2) = 58: 11224433, 11244332, 11332244, 11332442, 11334422, 11344322, ..., 44112233, 44112332, 44122133, 44122331, 44123321, 44133122.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270.

Alois P. Heinz, Table of n, a(n) for n = 0..206
Wikipedia, Eulerian numbers of the second kind

Cf. A008517, A201637, A112007, A163936, A288874.

a:= n-> combinat[eulerian2](2*n, n):
seq(a(n), n=0..20);
# second Maple program:
b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
     `if`(n=0, 1, (2*n-k-1)*b(n-1, k-1)+(k+1)*b(n-1, k)))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..20);

b[n_, k_]:=b[n, k]=If[k<0 || k>n, 0, If[n==0, 1, (2*n - k  - 1)*b[n - 1, k - 1] + (k + 1)*b[n - 1, k]]]; Table[b[2n, n], {n, 0, 20}] (* Indranil Ghosh, Jul 27 2017, after second Maple program *)
Flatten[{1, Table[Sum[(-1)^(n-k) * Binomial[4*n + 1, n - k] * StirlingS1[2*n + k, k], {k, 1, n}], {n, 1, 15}]}] (* Vaclav Kotesovec, Aug 11 2018 *)

a(n) = A201637(2n,n) = A288874(2n,n) = <<2n,n>>, with <<.,.>> = second order Eulerian numbers or Eulerian numbers of the second kind.

a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.6899458441572699524424834032837129180107588318196320162637478870996171397... and c = 3.5414537300298411499842602111667139605122817390785452902057395704515855797... - Vaclav Kotesovec, Aug 11 2018

A290595 Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0.

Original entry on oeis.org

1, 1, 2, 4, 19, 4, 28, 222, 147, 8, 280, 3194, 4128, 887, 16, 3640, 55024, 113566, 52538, 4835, 32, 58240, 1107336, 3268788, 2562676, 555684, 25167, 64, 1106560, 25526192, 100544412, 117517960, 45415640, 5301150, 128203, 128, 24344320, 663605680, 3325767376, 5352311764, 3189383200, 695714590, 47537320, 646519, 256, 608608000, 19213911360, 118361719296, 248493947496, 208996478388, 72479948400, 9696965250, 410038434, 3245139, 512

Offset: 0

Wolfdieter Lang, Aug 08 2017

The ordinary generating function (o.g.f.) of the (n+1)-th diagonal sequence of the Sheffer triangle A286718 = ((1 - 3*x)^(-1/3), -log(1 - 3*x)/3), called |S1hat[3,1]|, is GD(3,1;n,x) = P(n, x)/(1 - x)^(2*n+1), with the row polynomials P(n, x) = Sum_{k=0..n} T(n, k)*x^k, n >= 0.

For the two parameter Sheffer case |S1hat[d,a]| = ((1 - d*x)^{-a/d}, -log(1 - d*x)/d) (with gcd(d,a) = 1, d >=0, a >= 0, and for d = 1 one takes a = 0) the e.g.f. ED(t, x) of the o.g.f.s {GD(d,a;n,x)}_{n>=0} of the diagonal sequences with elements D(d,a;n,m) = |S1hat[d,a]|(n+m, m) (n=0 for the main diagonal) is of interest. It can be computed via Lagrange's theorem. For the special Sheffer case (1, f(x)) this has been done by P. Bala (see the link). This method can be generalized for Sheffer (g(x), f(x)), as shown in the W. Lang link.

			The triangle T(n, k) begins:
n\k        0        1         2         3        4       5      6   7 ...
0:         1
1:         1        2
2:         4       19         4
3:        28      222       147         8
4:       280     3194      4128       887       16
5:      3640    55024    113566     52538     4835      32
6:     58240  1107336   3268788   2562676   555684   25167      6
7:   1106560 25526192 100544412 117517960 45415640 5301150 128203 128
...
n = 8: 24344320 663605680 3325767376 5352311764 3189383200 695714590 47537320 646519 256,
n = 9: 608608000 19213911360 118361719296 248493947496 208996478388 72479948400 9696965250 410038434 3245139 512.
n = 3: The o.g.f. of the 4th diagonal sequence of A286718, [28, 418, 2485, ...] = A024213(n+1), n >= 0, is P(3, x) = (28 + 222*x + 147*x^2 + 8*x^3)/(1 - 3*x)^7.

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
Wolfdieter Lang, On Generating functions of Diagonal Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.

Cf. A024213, A286718, A288875 ([2,1] case).

T(n, k) = [x^k] P(n, x) with the numerator polynomials of the o.g.f. GD(n, x) = P(n, x)/(1-x)^(2*n+1) of the (n+1)-th diagonal sequence of the triangle A286718. See a comment above.

cp's OEIS Frontend

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A202017 Triangle of coefficients of the numerator polynomials of the rational o.g.f.'s of the diagonals of A059297.

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A288874 Row reversed version of triangle A201637 (second-order Eulerian triangle).

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A288875 Triangle read by rows. The rows give the coefficients of the numerator polynomials for the o.g.f.s of the diagonal sequences of triangle A028338.

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A290306 Number of permutations of the multiset {1,1,2,2,...,2n,2n} having exactly n ascents and no number smaller than k between the two occurrences of any number k.

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