A111578 -id:A111578 - cp's OEIS frontend

A048854 Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].

1, 2, 1, 12, 12, 1, 120, 180, 30, 1, 1680, 3360, 840, 56, 1, 30240, 75600, 25200, 2520, 90, 1, 665280, 1995840, 831600, 110880, 5940, 132, 1, 17297280, 60540480, 30270240, 5045040, 360360, 12012, 182, 1, 518918400, 2075673600, 1210809600, 242161920, 21621600, 960960, 21840, 240, 1, 17643225600, 79394515200, 52929676800, 12350257920, 1323241920, 73513440, 2227680, 36720, 306, 1

Offset: 0

Wolfdieter Lang

s(n,x) := Sum_{m=0..n} T(n,m)*x^m are monic polynomials satisfying s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), with polynomials p(n,x) = Sum_{m=1..n} A048786(n,m)*x^m (row polynomials of triangle A048786) and p(0,x)=1.
In the umbral calculus (see the Roman reference, p. 21) the s(n,x) are called Sheffer polynomials for(1/sqrt(1+4*t),t/(1+4*t)). Here the Sheffer notation differs. See the W. Lang link under A006232.
For the general L[d,a] triangles see A286724, also for references.
This is the generalized signless Lah number triangle L[4,1], the Sheffer triangle ((1 - 4*t)^(-1/2), t/(1 - 4*t)). It is defined as transition matrix
risefac[4,1](x, n) = Sum_{m=0..n} L[4,1](n, m)*fallfac[4,1](x, m), where risefac[4,1](x, n) := Product_{0..n-1} (x  + (1 + 4*j)) for n >= 1 and risefac[4,1](x, 0) := 1, and  fallfac[4,1](x, n) := Product_{0..n-1} (x - (1 + 4*j)) for n >= 1 and fallfac[4,1](x, 0) := 1.
In matrix notation: L[4,1] = S1phat[4,1]*S2hat[4,1] with the unsigned scaled Stirling1 and the scaled Stirling2 generalizations A290319 and A111578 (but here with offsets 0), respectively.
The a- and z-sequences for this Sheffer matrix have e.g.f.s Ea(t) = 1 + 4*t and Ez(t) = (1 + 4*t)*(1 - (1 + 4*t)^(-1/2))/t, respectively. That is, a = {1, 4, repeat(0)} and z(n) = 2*A292220(n). See the W. Lang link on a- and z-sequences there.
The inverse matrix T^(-1) = L^(-1)[4,1] is Sheffer ((1 + 4*t)^(-1/2), t/(1 + 4*t)). This means that T^(-1)(n, m) = (-1)^(n-m)*T(n, m).
fallfac[4,1](x, n) = Sum_{m=0..n} (-1)^(n-m)*T(n, m)*risefac[4,1](x, m), n >= 0.
Diagonal sequences have o.g.f. G(d, x) = A001813(d)*Sum_{m=0..d} A091042(d, m)*x^m/(1 - x)^{2*d + 1}, for d >= 0 (d=0 main diagonal). G(d, x) generates {A001813(d)*binomial(2*(n + d),2*d)}{n >= 0}. See the second W. Lang link on how to compute o.g.f.s of diagonal sequences of general Sheffer triangles. - _Wolfdieter Lang, Oct 12 2017

			The triangle T(n, m) begins:
n\m         0          1          2         3        4      5     6   7 8  ...
0:          1
1:          2          1
2:         12         12          1
3:        120        180         30         1
4:       1680       3360        840        56        1
5:      30240      75600      25200      2520       90      1
6:     665280    1995840     831600    110880     5940    132     1
7:   17297280   60540480   30270240   5045040   360360  12012   182   1
8:  518918400 2075673600 1210809600 242161920 21621600 960960 21840 240 1
...
n = 9: 17643225600 79394515200 52929676800 12350257920 1323241920 73513440 2227680 36720 306 1,
n = 10: 670442572800 3352212864000 2514159648000 670442572800 83805321600 5587021440 211629600 4651200 58140 380 1.
...
Recurrence from a-sequence: T(4, 2) = 2*T(3, 1) + 4*4*T(3, 2) = 2*180 + 16*30 = 840.
Recurrence from z-sequence: T(4, 0) = 4*(z(0)*T(3, 0) + z(1)*T(3, 1) + z(2)*T(3, 2)+ z(3)*T(3, 3)) = 4*(2*120 + 2*180 - 8*30 + 60*1) = 1680.
Four term recurrence: T(4, 2) = T(3, 1) + 2*13*T(3, 2) - 8*3*5*T(2, 2) =  180 + 26*30 - 120*1 = 840.
Meixner type identity for n = 2: (D_x - 4*(D_x)^2)*(12 + 12*x + 1*x^2) = (12 + 2*x) - 4*2 = 2*(2 + x).
Sheffer recurrence for R(3, x): [(2 + x) + 8*(1 + x)*D_x + 16*x*(D_x)^2] (12 + 12*x + 1*x^2) = (2 + x)*(12 + 12*x + x^2) + 8*(1 + x)*(12 + 2*x) + 16*2*x = 120 + 180*x + 30*x^2 + x^3 = R(3, x).
Boas-Buck recurrence for column m = 2 with n = 4: T(4, 2) = (4!*10/2)*(1*30/3! + 4*1/2!) = 840.
Diagonal sequence d = 2: {12, 180, 840 ...} has o.g.f. 12*(1 + 10*x + 5*x^2)/(1 - x)^5  (see A001813(2) and row n=2 of A091042) generating
{12*binomial(2*(n + 2), 4)}_{n >= 0}. - _Wolfdieter Lang_, Oct 12 2017

S. Roman, The Umbral Calculus, Academic Press, New York, 1984.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Orli Herscovici, Ross G. Pinsky, An identity involving stirling numbers of both kinds and its connection to right-to-left minima of certain set partitions, (2019).
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017, section C) 4.
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.

Related to triangle A046521. Cf. A048786. a(n, 0) = A001813.
A111578, A271703 L[1,0], A286724 L[2,1], A290319, A290596 L[3,1], A290597 L[3,2], A292220.
The diagonal sequences are: A000012, 2*A000384(n+1), 12*A053134, 120*A053135, 1680*A053137, ... - Wolfdieter Lang, Oct 12 2017

A290604_row := proc(n) exp(x*t/(1-4*t))/sqrt(1-4*t): series(%, t, n+2): seq(n!*coeff(coeff(%,t,n),x,j), j=0..n) end: seq(A290604_row(n), n=0..9); # Peter Luschny, Sep 23 2017

T[n_, m_] := n!/m! * Binomial[2*n, n] * Binomial[n, m] / Binomial[2*m, m]; Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013 *)
T[0, 0] = 1; T[-1, ] = T[, -1] = 0; T[n_, m_] /; n < m = 0; T[n_, m_] := T[n, m] = T[n-1, m-1] + 2*(4*n-3)*T[n-1, m] - 8*(n-1)*(2*n-3)*T[n-2, m]; Table[T[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Sep 23 2017 *)

T(n, m) = (n!/m!)*A046521(n, m) = (n!/m!)* binomial(2*n, n)*binomial(n, m)/binomial(2*m, m), n >= m >= 0, a(n, m) := 0, n < m.
Sum_{n>=0, k>=0} T(n, k)*x^n*y^k/(2*n)! = exp(x)*cosh(sqrt(x*y)). - Vladeta Jovovic, Feb 21 2003
T(n, m) = L[4,1](n,m) = Sum_{k=m..n} A290319(n, k)*A111578(k+1, m+1), 0 <= m <= n.
E.g.f. of row polynomials R(n, x) := Sum_{m=0..n} T(n, m)*x^m:
(1 - 4*t)^(-1/2)*exp(x*t/(1 - 4*t)) (this is the e.g.f. for the triangle).
E.g.f. of column m: (1 - 4*t)^(-1/2)*(t/(1 - 4*t))^m/m!, m >= 0.
Three term recurrence for column entries m >= 1: T(n, m) = (n/m)*T(n-1, m-1) + 4*n*T(n-1, m) with T(n, m) = 0 for n < m, and for the column m = 0: T(n, 0) = n*Sum_{j=0..n-1} z(j)*T(n-1, j), n >= 1, T(0, 0) = 0, from the a-sequence {1, 4 repeat(0)} and the z(j) = 2*A292220(j) (see above).
Four term recurrence: T(n, m) = T(n-1, m-1) + 2*(4*n - 3)*T(n-1, m) - 8*(n-1)*(2*n - 3)*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.
Meixner type identity for (monic) row polynomials: (D_x/(1 + 4*D_x)) * R(n, x) = n * R(n-1, x), n >= 1, with R(0, x) = 1 and D_x = d/dx. That is, Sum_{k=0..n-1} (-4)^k*(D_x)^(k+1)*R(n, x) = n*R(n-1, x), n >= 1.
General recurrence for Sheffer row polynomials (see the Roman reference, p. 50, Corollary 3.7.2, rewritten for the present Sheffer notation):
R(n, x) = [(2 + x)*1 + 8*(1 + x)*D_x + 16*x*(D_x)^2]*R(n-1, x), n >= 1, with R(0, x) = 1.
Boas-Buck recurrence for column m (see a comment in A286724 with references): T(n, m) = (n!/(n-m))*(2 + 4*m)*Sum_{p=0..n-1-m} 4^p*T(n-1-p, m)/(n-1-p)!, for n > m >= 0, with input T(m, m) = 1.
Explicit form (from the o.g.f.s of diagonal sequences): ((2*(n-m))!/(n-m)!)*binomial(2*n,2*(n-m)), n >= m >= 0, and vanishing for n < m. - Wolfdieter Lang, Oct 12 2017

Name changed, after merging my newer duplicate, from Wolfdieter Lang, Oct 10 2017

A286718 Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3x)^(-1/3), (-1/3)log(1 - 3*x)). A generalized Stirling1 triangle.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 28, 39, 12, 1, 280, 418, 159, 22, 1, 3640, 5714, 2485, 445, 35, 1, 58240, 95064, 45474, 9605, 1005, 51, 1, 1106560, 1864456, 959070, 227969, 28700, 1974, 70, 1, 24344320, 42124592, 22963996, 5974388, 859369, 72128, 3514, 92, 1, 608608000, 1077459120, 616224492, 172323696, 27458613, 2662569, 159978, 5814, 117, 1, 17041024000, 30777463360, 18331744896, 5441287980, 941164860, 102010545, 7141953, 322770, 9090, 145, 1

Offset: 0

Wolfdieter Lang, May 18 2017

This is a generalization of the unsigned Stirling1 triangle A132393.
In general the lower triangular Sheffer matrix ((1 - d*x)^(-a/d), (-1/d)*log(1 - d*x)) is called here |S1hat[d,a]|. The signed matrix S1hat[d,a] with elements (-1)^(n-k)*|S1hat[d,a]|(n, k) is the inverse of the generalized Stirling2 Sheffer matrix S2hat[d,a] with elements S2[d,a](n, k)/d^k, where S2[d,a] is Sheffer (exp(a*x), exp(d*x) - 1).
In the Bala link the signed S1hat[d,a] (with row scaled elements S1[d,a](n,k)/d^n where S1[d,a] is the inverse matrix of S2[d,a]) is denoted by s_{(d,0,a)}, and there the notion exponential Riordan array is used for Sheffer array.
In the Luschny link the elements of |S1hat[m,m-1]| are called Stirling-Frobenius cycle numbers SF-C with parameter m.
From Wolfdieter Lang, Aug 09 2017: (Start)
The general row polynomials R(d,a;n,x) = Sum_{k=0..n} T(d,a;n,k)*x^k of the Sheffer triangle |S1hat[d,a]| satisfy, as special polynomials of the Boas-Buck class (see the reference), the identity (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!*Sum_{k=0..n-1} d^k*(a*1 + d*beta(k)*E_x)*R(d,a;n-1-k,x)/(n-1-k)!, for n >= 0, with E_x = x*d/dx (Euler operator), and beta(k) = A002208(k+1)/A002209(k+1).
This entails a recurrence for the sequence of column k, for n > k >= 0: T(d,a;n,k) = (n!/(n - k))*Sum_{p=k..n-1} d^(n-1-p)*(a + d*k*beta(n-1-p))*T(d,a;p,k)/p!, with input T(d,a;k,k) = 1. For the present [d,a] = [3,1] case see the formula and example sections below. (End)
The inverse of the Sheffer triangular matrix S2[3,1] = A282629 is the Sheffer matrix S1[3,1] = (1/(1 + x)^(1/3), log(1 + x)/3) with rational elements S1[3,1](n, k) = (-1)^(n-m)*T(n, k)/3^n. - Wolfdieter Lang, Nov 15 2018

			The triangle T(n, k) begins:
n\k        0        1        2       3      4     5    6  7 8 ...
O:         1
1:         1        1
2:         4        5        1
3:        28       39       12       1
4:       280      418      159      22      1
5:      3640     5714     2485     445     35     1
6:     58240    95064    45474    9605   1005    51    1
7:   1106560  1864456   959070  227969  28700  1974   70  1
8:  24344320 42124592 22963996 5974388 859369 72128 3514 92 1
...
From _Wolfdieter Lang_, Aug 09 2017: (Start)
Recurrence: T(3, 1) = T(2, 0) + (3*3-2)*T(2, 1) = 4 + 7*5 = 39.
Boas-Buck recurrence for column k = 2 and n = 5:
T(5, 2) = (5!/3)*(3^2*(1 + 6*(3/8))*T(2,2)/2! + 3*(1 + 6*(5/12)*T(3, 2)/3! + (1 + 6*(1/2))* T(4, 2)/4!)) = (5!/3)*(9*(1 + 9/4)/2 + 3*(1 + 15/6)*12/6 + (1 + 3)*159/24) = 2485.
The beta sequence begins: {1/2, 5/12, 3/8, 251/720, 95/288, 19087/60480, ...}.
(End)

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.

P. Bala, A 3 parameter family of generalized Stirling numbers
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 Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Peter Luschny, The Stirling-Frobenius numbers.

S2[d,a] for [d,a] = [1,0], [2,1], [3,1], [3,2], [4,1] and [4,3] is A048993, A154537, A282629, A225466, A285061 and A225467, respectively.
S2hat[d,a] for these [d,a] values is A048993, A039755, A111577 (offset 0), A225468, A111578 (offset 0) and A225469, respectively.
|S1hat[d,a]| for [d,a] = [1,0], [2,1], [3,2], [4,1] and [4,3] is A132393, A028338, A225470, A290317 and A225471, respectively.
Column sequences for k = 0..4: A007559, A024216(n-1), A286721(n-2), A382984, A382985.
Diagonal sequences: A000012, A000326(n+1), A024212(n+1), A024213(n+1).
Row sums: A008544. Alternating row sums: A000007.
Beta sequence: A002208(n+1)/A002209(n+1).

T[n_ /; n >= 1, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + (3*n-2)* T[n-1, k]; T[, -1] = 0; T[0, 0] = 1; T[n, k_] /; nJean-François Alcover, Jun 20 2018 *)

Recurrence: T(n, k) = T(n-1, k-1) + (3*n-2)*T(n-1, k), for n >= 1, k = 0..n, and T(n, -1) = 0, T(0, 0) = 1 and T(n, k) = 0 for n < k.
E.g.f. of row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k (i.e., e.g.f. of the triangle) is (1 - 3*z)^{-(x+1)/3}.
E.g.f. of column k is (1 - 3*x)^(-1/3)*((-1/3)*log(1 - 3*x))^k/k!.
Recurrence for row polynomials is R(n, x) = (x+1)*R(n-1, x+3), with R(0, x) = 1.
Row polynomial R(n, x) = risefac(3,1;x,n) with the rising factorial
risefac(d,a;x,n) := Product_{j=0..n-1} (x + (a + j*d)). (For the signed case see the Bala link, eq. (16)).
T(n, k) = sigma^{(n)}{n-k}(a_0,a_1,...,a{n-1}) with the elementary symmetric functions with indeterminates a_j = 1 + 3*j.
T(n, k) = Sum_{j=0..n-k} binomial(n-j, k)*|S1|(n, n-j)*3^j, with the unsigned Stirling1 triangle |S1| = A132393.
Boas-Buck column recurrence (see a comment above): T(n, k) =
(n!/(n - k))*Sum_{p=k..n-1} 3^(n-1-p)*(1 + 3*k*beta(n-1-p))*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, with beta(k) = A002208(k+1)/A002209(k+1). See an example below. - Wolfdieter Lang, Aug 09 2017

A285061 Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).

Original entry on oeis.org

1, 1, 4, 1, 24, 16, 1, 124, 240, 64, 1, 624, 2656, 1792, 256, 1, 3124, 26400, 33920, 11520, 1024, 1, 15624, 250096, 546560, 331520, 67584, 4096, 1, 78124, 2313360, 8105664, 7822080, 2745344, 372736, 16384, 1, 390624, 21132736, 114627072, 165398016, 88940544, 20299776, 1966080, 65536, 1, 1953124, 191757120, 1574682880, 3270274560, 2529343488, 863256576, 138215424, 10027008, 262144

Offset: 0

Wolfdieter Lang, Apr 13 2017

For Sheffer triangles (infinite lower triangular exponential convolution matrices) see the W. Lang link under A006232, with references.
This is a generalization of the Sheffer triangle Stirling2(n, m) = A048993(n, m) denoted by (exp(x), exp(x)-1), which could be named S2[1,0].
For the Sheffer triangle (exp(x), (1/4)*(exp(4*x) - 1)) see A111578, also the P. Bala link where this triangle, T(n, m)/4^m, is named S_{(4,0,1)}. The triangle T(n, m)*m! is given in A285066.
The a-sequence for this Sheffer triangle has e.g.f. 4*x/log(1+x) and is 4*A006232(n)/A006233(n) (Cauchy numbers of the first kind).
The z-sequence has e.g.f. (4/(log(1+x)))*(1 - 1/(1+x)^(1/4)) and is A285062(n)/A285063(n).
The main diagonal gives A000302.
The row sums give A285064. The alternating row sums give A285065.
The first column sequences are A000012, 4*A003463, 4^2*A016234. For the e.g.f.s and o.g.f.s see the formula section.
This triangle appears in the o.g.f. G(n, x) of the sequence {(1 + 4*m)^n}{m>=0}, as G(n, x) = Sum{m=0..n} T(n, m)*m!*x^m/(1-x)^(m+1), n >= 0. Hence the corresponding e.g.f. is, by the linear inverse Laplace transform, E(n, t) = Sum_{m >=0}(1 + 4*m)^n t^m/m! = exp(t)*Sum_{m=0..n} T(n, m)*t^m.
The corresponding Euler triangle with reversed rows is rEu(n, k) = Sum_{m=0..k} (-1)^(k-m)*binomial(n-m, k-m)*T(n, k)*k!, 0 <= k <= n. This is A225118 with row reversion.
In general the Sheffer triangle S2[d,a] appears in the reordering of the operator (a*1 + d*E_x) = Sum_{m=0..n} S2(d,a;n,m) x^m*(d_x)^m with the derivative d_x and the Euler operator E_x = x*d_x. For [d,a] = [1,0] this becomes the standard Stirling2 property.

			The triangle T(n,m) begins:
  n\m  0      1        2         3         4        6        7       8     9
  0:   1
  1:   1      4
  2:   1     24       16
  3:   1    124      240        64
  4:   1    624     2656      1792       256
  5:   1   3124    26400     33920     11520     1024
  6:   1  15624   250096    546560    331520    67584     4096
  7:   1  78124  2313360   8105664   7822080  2745344   372736   16384
  8:   1 390624 21132736 114627072 165398016 88940544 20299776 1966080 65536
  ...
Three term recurrence: T(4, 1) = 4*1 + (1 + 4*1)*124 = 624.
Recurrence for row polynomial R(3, x) (Meixner type): ((1 + 4*x) + 4*x*d_x)*(1 + 24*x + 16*x^2) = 1 + 124*x + 240*x^2 + 64*x^3.
Boas-Buck recurrence for column m = 2, and n = 4: T(4, 2) = (1/2)*(2*(2 + 4*2)*T(3, 2) + 2*6*(-4)^2*Bernoulli(2)*T(2, 2)) = (1/2)*(20*240 + 12*16*(1/6)*16) = 2656. - _Wolfdieter Lang_, Aug 11 2017

Michael De Vlieger, Table of n, a(n) for n = 0..11475, rows n = 0..150, flattened.
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 9.
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.

Cf. A000012, A000302, 4*A003463, A111578, A006232/A006233, 16*A016234, A282629, A285062/A285063, A285064, A285065, A285066.

Table[Sum[Binomial[n, k]*4^k*StirlingS2[k, m], {k, 0, n}], {n, 0, 20}, {m, 0, n}] // Flatten (* Indranil Ghosh, May 06 2017 *)

T(n, m) = sum(k=0, n, binomial(n, k)*4^k*stirling(k, m, 2));
for(n=0, 20, for(m=0, n, print1(T(n, m),", ");); print();) \\ Indranil Ghosh, May 06 2017

Three term recurrence (from the conversion property mentioned above, with [d,a] =[4,1]): T(n, -1) = 0, T(0, 0) = 1, T(n, m) = 0 if n < m. T(n, m) = 4*T(n-1, m-1) + (1 + 4*m)*T(n-1, m) for n >= 1, m = 0..n.
T(n, m) = Sum_{k=0..m} binomial(m,k)*(-1)^(k-m)*(1 + 4*k)^n/m!, 0 <= m <= n, satisfying the recurrence.
E.g.f. of the row polynomials R(n, x) = Sum_{m=0..n} T(n, m)*x^m: exp(z)*exp(x*(exp(4*z) - 1)). This is the e.g.f. of the triangle.
E.g.f. for the sequence of column m: exp(x)*((exp(3*x) - 1)^m)/m! (Sheffer property).
O.g.f. for sequence of column m: (4*x)^m/Product_{j=0..m} (1 - (1 + 4*j)*x) (by Laplace transform of the e.g.f.).
T(n, m) = Sum_{k=0..n} binomial(n, k)* 4^k*Stirling2(k, m), 0 <= m <= n, where Stirling2 is given in A048993.
A nontrivial recurrence for the column m=0 entries T(n, 0) = 4^n from the z-sequence given above: T(n,0) = n*Sum_{j=0..n-1} z(j)*T(n-1,j), n >= 1, T(0, 0) = 1.
Recurrence for column m >= 1 entries from the a-sequence given above: T(n, m) = (n/m)* Sum_{j=0..n-m} binomial(m-1+j, m-1)*a(j)*T(n-1, m-1+j), m >= 1.
Recurrence for row polynomials R(n, x) (Meixner type): R(n, x) = ((1 + 4*x) + 4*x*d_x)*R(n-1, x), with differentiation d_x, for n >= 1, with input R(0, x) = 1.
Boas-Buck recurrence for column sequence m: T(n, m) = (1/(n - m))*((n/2)*(2 + 4*m)*T(n-1, m) + m*Sum_{p=m..n-2} binomial(n, p)*(-4)^(n-p)*Bernoulli(n-p)*T(p, m)), for n > m >= 0, with input T(m,m) = 4^m. See a comment and references in A282629. An example is given below. - Wolfdieter Lang, Aug 11 2017

A111669 Triangle read by rows, based on a simple Fibonacci recursion rule.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 11, 7, 1, 1, 5, 26, 32, 12, 1, 1, 6, 57, 122, 92, 20, 1, 1, 7, 120, 423, 582, 252, 33, 1, 1, 8, 247, 1389, 3333, 2598, 681, 54, 1, 1, 9, 502, 4414, 18054, 24117, 11451, 1815, 88, 1, 1, 10, 1013, 13744, 94684, 210990, 172980, 49566, 4807, 143, 1

Offset: 0

Gary W. Adamson, Aug 14 2005

Subdiagonal is A000071(n+3). Row sums of inverse are 0^n.

Row sums are given by A135934. - Emanuele Munarini, Dec 05 2017

			Triangle begins
  1....1....2....3....5....8...13....F(k+1)
  1
  1....1
  1....2....1
  1....3....4....1
  1....4...11....7....1
  1....5...26...32...12....1
  1....6...57..122...92...20....1
For example, T(6,3) = 122 = 26 + 3*32 = T(5,2) + F(4)*T(5,3).

Michel Marcus, Rows n = 0..20, flattened

Cf. A111577, A111578, A111579, A008277, A039755, A135934.

(* To generate the triangle *)
Grid[RecurrenceTable[{F[n,k] == F[n-1,k-1] + Fibonacci[k+1] F[n-1,k], F[0,k] == KroneckerDelta[k]}, F, {n,0,10}, {k,0,10}]] (* Emanuele Munarini, Dec 05 2017 *)

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, T(n-1, k-1) + fibonacci(k+1)*T(n-1, k))); \\ Michel Marcus, May 25 2024

T(n, k) = T(n-1, k-1) + F(k+1)*T(n-1, k) where F(n)=A000045(n).

Column k has g.f. x^k/Product_{j=0..k} (1 - F(j+1)*x).

Edited by Paul Barry, Nov 14 2005

A111579 Triangle A(r,c) read by rows, which contains the row sums of the triangle T(n,k)= T(n-1,k-1)+((c-1)k+1)T(n-1,k) in column c.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 5, 2, 1, 1, 16, 15, 6, 2, 1, 1, 32, 52, 24, 7, 2, 1, 1, 64, 203, 116, 35, 8, 2, 1, 1, 128, 877, 648, 214, 48, 9, 2, 1, 1, 256, 4140, 4088, 1523, 352, 63, 10, 2, 1, 1, 512, 21147, 28640, 12349, 3008, 536, 80, 11, 2, 1

Offset: 0

Gary W. Adamson, Aug 07 2005

Triangles of generalized Stirling numbers of the second kind may be defined by recurrences T(n,k) = T(n-1,k-1) + Q*T(n-1,k) initialized by T(0,0)=T(1,0)=T(1,1)=1. Q=1 generates Pascal's triangle A007318,
Q=k+1 generates A008277, Q=2k+1 generates A039755, Q=3k+1 generates A111577, Q=4k+1 generates A111578, Q=5k+1 generates A166973.
(These definitions assume row and column enumeration 0<=n, 0<=k<=n.)
Each of these triangles characterized by Q=(c-1)*k+1 has row sums sum_{k=0..n} T(n,k), which define the column A(.,c).

Cf. A008277, A000110, A039755, A004211, A111577, A111578.

T := proc(n,k,c) if k < 0 or k > n then 0 ; elif n <= 1 then 1; else procname(n-1,k-1,c)+((c-1)*k+1)*procname(n-1,k,c) ; fi; end:
A111579 := proc(r,c) local n; if c = 0 then 1 ; else n := r-c ; add( T(n,k,c),k=0..n) ; end if; end:
seq(seq(A111579(r,c),c=0..r),r=0..10) ; # R. J. Mathar, Oct 30 2009

T[n_, k_, c_] := T[n, k, c] = If[k < 0 || k > n, 0, If[n <= 1, 1, T[n-1, k-1, c] + ((c-1)*k+1)*T[n-1, k, c]]];
A111579[r_, c_] := Module[{n}, If[c == 0, 1, n = r - c; Sum[T[n, k, c], {k, 0, n}]]];
Table[A111579[r, c], {r, 0, 10}, {c, 0, r}] // Flatten (* Jean-François Alcover, Aug 01 2023, after R. J. Mathar *)

A(r=n+c,c) = sum_{k=0..n} T(n,k,c), 0<=c<=r where T(n,k,c) = T(n-1,k-1,c) + ((c-1)*k+1)*T(n-1,k,c).
A(r,0) = 1.
A(r,1) = 2^(r-1).
A(r,2) = A000110(r-1).
A(r,3) = A007405(r-3).

Edited by R. J. Mathar, Oct 30 2009

A285066 Triangle read by rows: T(n, m) = A285061(n, m)*m!, 0 <= m <= n.

Original entry on oeis.org

1, 1, 4, 1, 24, 32, 1, 124, 480, 384, 1, 624, 5312, 10752, 6144, 1, 3124, 52800, 203520, 276480, 122880, 1, 15624, 500192, 3279360, 7956480, 8110080, 2949120, 1, 78124, 4626720, 48633984, 187729920, 329441280, 268369920, 82575360, 1, 390624, 42265472, 687762432, 3969552384, 10672865280, 14615838720, 9909043200, 2642411520, 1, 1953124, 383514240, 9448097280, 78486589440, 303521218560, 621544734720, 696605736960, 404288962560, 95126814720

Offset: 0

Wolfdieter Lang, Apr 19 2017

This is the Sheffer triangle S2[4,1] = A285061 with column m scaled by m!. This is the fourth member of the triangle family A131689, A145901 and A284861.
This triangle appears in the o.g.f. G(n, x) = Sum_{m=0..n} T(n, m)*x^m/(1-x)^(m+1), n >= 0, of the power sequence {(1+4*m)^n}_{m >= 0}.
The diagonal sequence is A047053. The row sums give A285067. The alternating sum of row n is A141413(n+2), n >= 0.
The first column sequences are: A000012, 4*A003463, 2!*4^2*A016234.

			The triangle T(n, m) begins:
  n\m 0     1       2        3         4         5         6        7
  0:  1
  1:  1     4
  2:  1    24      32
  3:  1   124     480      384
  4:  1   624    5312    10752      6144
  5:  1  3124   52800   203520    276480    122880
  6:  1 15624  500192  3279360   7956480   8110080   2949120
  7:  1 78124 4626720 48633984 187729920 329441280 268369920 82575360
  ...
row 8: 1 390624 42265472 687762432 3969552384 10672865280 14615838720 9909043200 2642411520
row 9: 1 1953124 383514240 9448097280 78486589440 303521218560 621544734720 696605736960 404288962560 95126814720
...

Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.

Cf. A000012, 4*A003463, 32*A016234, A111578 (T(n, m)/4^m), A131689, A141413, A145901, A225118, A225472, A225473, A284861, A285061.

T[n_, m_]:=Sum[Binomial[m, k]*(-1)^(k - m)*(1 + 4k)^n, {k, 0, n}]; Table[T[n, m], {n, 0, 10},{m, 0, n}] // Flatten (* Indranil Ghosh, May 02 2017 *)

from sympy import binomial
def T(n, m):
    return sum([binomial(m, k)*(-1)**(k - m)*(1 + 4*k)**n for k in range(n + 1)])
for n in range(21):
    print([T(n, m) for m in range(n + 1)])
# Indranil Ghosh, May 02 2017

T(n, m) = A285061(n, m)*m! = A111578(n, m)*(4^m*m!), 0 <= m <= n.
T(n, m) = Sum_{k=0..n} binomial(m,k)*(-1)^(k-m)*(1+4*k)^n.
T(n, m) = Sum_{j=0..n} binomial(n-j,m-j)*A225118(n,n-j).
Recurrence: T(n, -1) = 0, T(0, 0) = 1, T(n, m) = 0 if n < m and T(n, m) =
4*m*T(n-1, m-1) + (1+4*m)*T(n-1, m) for n >= 1, m=0..n.
E.g.f. row polynomials R(n, x) = Sum_{m=0..n} T(n, m)*x^m: exp(z)/(1 - x*(exp(4*z) - 1)).
E.g.f. column m: exp(x)*(exp(4*x) - 1)^m, m >= 0.
O.g.f. column m: m!*(4*x)^m/Product_{j=0..m} (1 - (1 + 4*j)*x), m >= 0.

A166973 Triangle T(n,k) read by rows: T(n, k) = (mn - mk + 1)T(n - 1, k - 1) + (5k - 4)(mk - (m - 1))*T(n - 1, k) where m = 0.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 43, 18, 1, 1, 259, 241, 34, 1, 1, 1555, 2910, 785, 55, 1, 1, 9331, 33565, 15470, 1940, 81, 1, 1, 55987, 378546, 281085, 56210, 4046, 112, 1, 1, 335923, 4219993, 4875906, 1461495, 161406, 7518, 148, 1, 1, 2015539, 46755846, 82234489

Offset: 1

Roger L. Bagula, Oct 26 2009

The recursion T(n, k) = (m*n - m*k + 1)*T(n-1, k-1) + (5*k - 4)*(m*k - (m - 1))*T(n-1, k) was intended to range over m values 0 to 4 as given by the original Mathematica code. This sequences is the case for m = 0. - G. C. Greubel, May 29 2016

With offset 0 in the rows and columns this is the Sheffer triangle S2[5,1] = (exp(x), (exp(5*x) - 1)/5). See S2[4,1] = A111578 (with offsets 0), S[3,1] = A111577 (with offsets 0), S2[2,1] = A039755

			Triangle T(n, k) starts:
n\k   1       2        3        4        5       6      7     8   9 10 ...
1:    1
2:    1       1
3:    1       7        1
4:    1      43       18        1
5:    1     259      241       34        1
6:    1    1555     2910      785       55       1
7:    1    9331    33565    15470     1940      81      1
8:    1   55987   378546   281085    56210    4046    112     1
9:    1  335923  4219993  4875906  1461495  161406   7518   148   1
10:   1 2015539 46755846 82234489 35567301 5658051 394464 12846 189  1
... Reformatted, - _Wolfdieter Lang_, Aug 13 2017

G. C. Greubel, Table of n, a(n) for the first 25 rows

Cf. A111577.

S2[4,1] = A111578 (with offsets 0), S2[3,1] = A111577 (with offsets 0), S2[2,1] = A039755. - Wolfdieter Lang, Aug 13 2017

A[n_, 1] := 1; A[n_, n_] := 1; A[n_, k_] := A[n - 1, k - 1] + (5*k - 4)*A[n - 1,k]; Flatten[ Table[A[n, k], {n, 10}, {k, n}]] (* modified by G. C. Greubel, May 29 2016 *)

T(n, k) = T(n - 1, k - 1) + (5*k - 4)*T(n - 1, k).

E.g.f. column k: int(exp(x)*((exp(5*x)-1)/5)^(k-1)/(k-1)!, x) + (-1)^k/A008548(k). - Wolfdieter Lang, Aug 13 2017

cp's OEIS Frontend

A048854 Triangle read by rows. A generalization of unsigned Lah numbers, called L[4,1].

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A286718 Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A285061 Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A111669 Triangle read by rows, based on a simple Fibonacci recursion rule.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A111579 Triangle A(r,c) read by rows, which contains the row sums of the triangle T(n,k)= T(n-1,k-1)+((c-1)*k+1)*T(n-1,k) in column c.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A285066 Triangle read by rows: T(n, m) = A285061(n, m)*m!, 0 <= m <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A166973 Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.

Views

Author

Keywords

Comments

A286718 Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3x)^(-1/3), (-1/3)log(1 - 3*x)). A generalized Stirling1 triangle.

A111579 Triangle A(r,c) read by rows, which contains the row sums of the triangle T(n,k)= T(n-1,k-1)+((c-1)k+1)T(n-1,k) in column c.

A166973 Triangle T(n,k) read by rows: T(n, k) = (mn - mk + 1)T(n - 1, k - 1) + (5k - 4)(mk - (m - 1))*T(n - 1, k) where m = 0.