A075499 -id:A075499 - cp's OEIS frontend

A075907 Fourth column of triangle A075499.

1, 40, 1040, 22400, 435456, 7956480, 139694080, 2387968000, 40075329536, 663887544320, 10896534405120, 177653730508800, 2882307270639616, 46596186764738560, 751299029274460160, 12089975328525516800

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..3} A075513(4,m)*exp(4*(m+1)*x)/3!.

Index entries for linear recurrences with constant coefficients, signature (40, -560, 3200, -6144).

Cf. A019677, A075908.

Table[(-4^n+24*8^n-81*12^n+64*16^n)/6,{n,0,20}] (* or *) LinearRecurrence[ {40,-560,3200,-6144},{1,40,1040,22400},20] (* Harvey P. Dale, Jun 04 2013 *)

a(n) = A075499(n+4, 4) = (4^n)*S2(n+4, 4) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (-4^n + 24*8^n - 81*12^n + 64*16^n)/3!.
G.f.: 1/Product_{k=1..4} (1 - 4*k*x).
E.g.f.: (d^4/dx^4)(((exp(4*x)-1)/4)^4)/4! = (-exp(4*x) + 24*exp(8*x) - 81*exp(12*x) + 64*exp(16*x))/3!.
a(0)=1, a(1)=40, a(2)=1040, a(3)=22400, a(n) = 40*a(n-1) - 560*a(n-2) + 3200*a(n-3) - 6144*a(n-4). - Harvey P. Dale, Jun 04 2013

A075908 Fifth column of triangle A075499.

Original entry on oeis.org

1, 60, 2240, 67200, 1779456, 43545600, 1010606080, 22600089600, 492077121536, 10505429975040, 221005133905920, 4597756408627200, 94837435443183616, 1943344895628410880, 39618196941842677760

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..4} A075513(5,m)*exp(4*(m+1)*x)/4!.

Cf. A075907, A075909.

a(n) = A075499(n+5, 5) = (4^n)*S2(n+5, 5) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..4} A075513(5, m)*((m+1)*4)^n/4!.
G.f.: 1/Product_{k=1..5} (1 - 4*k*x).
E.g.f.: (d^5/dx^5)(((exp(4*x)-1)/4)^5)/5! = (exp(4*x) - 64*exp(8*x) + 486*exp(12*x) - 1024*exp(16*x) + 625*exp(20*x))/4!.

A075910 Seventh column of triangle A075499.

Original entry on oeis.org

1, 112, 7392, 376320, 16380672, 642453504, 23410376704, 808210923520, 26787271999488, 860325833342976, 26956901684084736, 828217683974553600, 25047119070415028224, 747831252926309859328, 22095179333791056396288

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..6} A075513(7,m)*exp(4*(m+1)*x)/6!.

Cf. A075509.

CoefficientList[Series[1/Product[1-4k x,{k,7}],{x,0,20}],x] (* Harvey P. Dale, Aug 11 2021 *)

a(n) = A075499(n+7, 7) = (4^n)*S2(n+7, 7) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..6} A075513(7, m)*((m+1)*4)^n/6!.
G.f.: 1/Product_{k=1..7} (1 - 4*k*x).
E.g.f.: (d^7/dx^7)(((exp(4*x)-1)/4)^7)/7! = (exp(4*x) - 384*exp(8*x) + 10935*exp(12*x) - 81920*exp(16*x) + 234375*exp(20*x) - 279936*exp(24*x) + 117649*exp(28*x))/6!.

A075909 Sixth column of triangle A075499.

Original entry on oeis.org

1, 84, 4256, 169344, 5843712, 183794688, 5421678592, 152720375808, 4157366140928, 110282217357312, 2867778350481408, 73424436820180992, 1857023919127527424, 46511918954689069056, 1155904251854380335104

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..5} A075513(6,m)*exp(4*(m+1)*x)/5!.

Cf. A075908, A075910.

a(n) = A075499(n+6, 6) = (4^n)*S2(n+6, 6) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..5} A075513(6, m)*((m+1)*4)^n/5!.
G.f.: 1/Product_{k=1..6} (1 - 4*k*x).
E.g.f.: (d^6/dx^6)(((exp(4*x)-1)/4)^6)/6! = (-exp(4*x) + 160*exp(8*x) - 2430*exp(12*x) + 10240*exp(16*x) - 15625*exp(20*x) + 7776*exp(24*x))/5!.

A099394 Triangle T(k,n) by rows: n! * A075499(k,n).

Original entry on oeis.org

1, 4, 1, 16, 12, 2, 64, 112, 48, 6, 256, 960, 800, 240, 24, 1024, 7936, 11520, 6240, 1440, 120, 4096, 64512, 154112, 134400, 53760, 10080, 720, 16384, 520192, 1978368, 2612736, 1612800, 510720, 80640, 5040, 65536, 4177920, 24780800

Offset: 0

Ralf Stephan, Oct 21 2004

Triangle given by [4,0,8,0,12,0,16,0,20,0,24,0,28,0,...] DELTA [1,1,2,2,3,3,4,4,5,5,6,6,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jun 04 2009

			Triangle begins:
     1;
     4,    1;
    16,   12,     2;
    64,  112,    48,    6;
   256,  960,   800,  240,   24;
  1024, 7936, 11520, 6240, 1440, 120;

Toufik Mansour, Generalization of some identities involving the Fibonacci numbers, arXiv:math/0301157 [math.CO], 2003.

T(n, k) = A028246(n+1, k+1)*4^(n-k) = Stirling2(n+1, k+1)*k!*4^(n-k), see A008277. - Philippe Deléham, Oct 02 2005

A004213 Shifts one place left under 4th-order binomial transform.

Original entry on oeis.org

1, 1, 5, 29, 201, 1657, 15821, 170389, 2032785, 26546673, 376085653, 5736591885, 93614616409, 1625661357673, 29905322979421, 580513190237573, 11850869542405409, 253669139947767777, 5678266212792053029, 132607996474971041789, 3224106929536557918697

Offset: 0

N. J. A. Sloane

Length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(k)<=F(k)+4 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=4, otherwise F(k+1)=F(k); see example and Fxtbook link. - Joerg Arndt, Apr 30 2011

			Restricted growth strings: a(0)=1 corresponds to the empty string, a(1)=1 to [0],
a(2)=3 to [00], [01], [02], [03], and [04], a(3) = 29 to
       RGS          F
.1:  [ 0 0 0 ]    [ 0 0 0 ]
.2:  [ 0 0 1 ]    [ 0 0 0 ]
.3:  [ 0 0 2 ]    [ 0 0 0 ]
.4:  [ 0 0 3 ]    [ 0 0 0 ]
.5:  [ 0 0 4 ]    [ 0 0 4 ]
.6:  [ 0 1 0 ]    [ 0 0 0 ]
.7:  [ 0 1 1 ]    [ 0 0 0 ]
.8:  [ 0 1 2 ]    [ 0 0 0 ]
.9:  [ 0 1 3 ]    [ 0 0 0 ]
10:  [ 0 1 4 ]    [ 0 0 4 ]
11:  [ 0 2 0 ]    [ 0 0 0 ]
12:  [ 0 2 1 ]    [ 0 0 0 ]
13:  [ 0 2 2 ]    [ 0 0 0 ]
14:  [ 0 2 3 ]    [ 0 0 0 ]
15:  [ 0 2 4 ]    [ 0 0 4 ]
16:  [ 0 3 0 ]    [ 0 0 0 ]
17:  [ 0 3 1 ]    [ 0 0 0 ]
18:  [ 0 3 2 ]    [ 0 0 0 ]
19:  [ 0 3 3 ]    [ 0 0 0 ]
20:  [ 0 3 4 ]    [ 0 0 4 ]
21:  [ 0 4 0 ]    [ 0 4 4 ]
22:  [ 0 4 1 ]    [ 0 4 4 ]
23:  [ 0 4 2 ]    [ 0 4 4 ]
24:  [ 0 4 3 ]    [ 0 4 4 ]
25:  [ 0 4 4 ]    [ 0 4 4 ]
26:  [ 0 4 5 ]    [ 0 4 4 ]
27:  [ 0 4 6 ]    [ 0 4 4 ]
28:  [ 0 4 7 ]    [ 0 4 4 ]
29:  [ 0 4 8 ]    [ 0 4 8 ]
[_Joerg Arndt_, Apr 30 2011]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..66
Joerg Arndt, Matters Computational (The Fxtbook), section 17.3.5, pp. 366-368
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
A. Kerber, A matrix of combinatorial numbers related to the symmetric groups<, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]
N. J. A. Sloane, Transforms

Cf. A075499 (row sums).

A004211 (RGS where s(k)<=F(k)+2), A004212 (s(k)<=F(k)+3), A005011 (s(k)<=F(k)+5), A000110 (s(k)<=F(k)+1). - Joerg Arndt, Apr 30 2011

A004213 := proc(n)
    add(4^(n-m)*combinat[stirling2](n,m),m=0..n) ;
end proc:
seq(A004213(n),n=0..30) ; # R. J. Mathar, Aug 20 2022

Table[4^n BellB[n, 1/4], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

a(n):=if n=0 then 1 else sum(4^(n-k)*binomial(n-1, k-1)*a(k-1), k, 1, n); /* Vladimir Kruchinin, Nov 28 2011 */

x='x+O('x^66);
egf=exp(intformal(exp(4*x))); /* =  1 + x + 5/2*x^2 + 29/6*x^3 + 67/8*x^4 + ... */
/* egf=exp(1/4*(exp(4*x)-1)) */ /* alternative computation */
Vec(serlaplace(egf)) /* Joerg Arndt, Apr 30 2011 */

a(n) = Sum_{m=0..n} 4^(n-m)*Stirling2(n, m).
E.g.f.: exp((exp(4*x)-1)/4).
O.g.f. A(x) satisfies A'(x)/A(x) = e^(4x).
E.g.f.: exp(Integral_{t = 0..x} exp(4*t)). - Joerg Arndt, Apr 30 2011
O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1-4*j*x). - Joerg Arndt, Apr 30 2011
Define f_1(x), f_2(x), ... such that f_1(x) = e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n = 2, 3, .... Then a(n) = e^(-1/4)*4^{n-1}*f_n(1/4). - Milan Janjic, May 30 2008
a(n) = upper left term in M^n, M = an infinite square production matrix in which a diagonal of (4,4,4,...) is appended to the right of Pascal's triangle:
  1, 4, 0, 0, 0, ...
  1, 1, 4, 0, 0, ...
  1, 2, 1, 4, 0, ...
  1, 3, 3, 1, 4, ...
  ... - Gary W. Adamson, Jul 29 2011
G.f. satisfies A(x) = 1 + x/(1 - 4*x)*A(x/(1 - 4*x)). a(n) = Sum_{k = 1..n} 4^(n-k)*binomial(n-1,k-1)*a(k-1), n > 0, a(0) = 1. - Vladimir Kruchinin, Nov 28 2011 [corrected by Ilya Gutkovskiy, May 02 2019]
G.f.: (G(0) - 1)/(x-1) where G(k) =  1 - 1/(1-4*k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 24 2013
G.f.: (G(0) - 1)/(1+x) where G(k) =  1 + 1/(1-4*k*x)/(1-x/(x+1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 31 2013
G.f.: T(0)/(1-x), where T(k) = 1 - 4*x^2*(k+1)/( 4*x^2*(k+1) - (1-x-4*x*k)*(1-5*x-4*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 19 2013
a(n) = exp(-1/4) * Sum_{k>=0} 4^(n-k) * k^n / k!. - Vaclav Kotesovec, Jul 15 2021
a(n) ~ 4^n * n^n * exp(n/LambertW(4*n) - 1/4 - n) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^n). - Vaclav Kotesovec, Jul 15 2021
From Peter Bala, Jun 29 2024: (Start)
a(n) = exp(-1/4)*Sum_{n >= 0} (4*n)^k/(n!*4^n).
Touchard's congruence holds: for odd prime p, a(p+k) == (a(k) + a(k+1)) (mod p) for k = 0,1,2,.... In particular, a(p) == 2 (mod p) for odd prime p. See A004211. (End)

A075498 Stirling2 triangle with scaled diagonals (powers of 3).

Original entry on oeis.org

1, 3, 1, 9, 9, 1, 27, 63, 18, 1, 81, 405, 225, 30, 1, 243, 2511, 2430, 585, 45, 1, 729, 15309, 24381, 9450, 1260, 63, 1, 2187, 92583, 234738, 137781, 28350, 2394, 84, 1, 6561, 557685, 2205225, 1888110, 563031, 71442, 4158, 108, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(3*z) - 1)*x/3) - 1.
Subtriangle of the triangle given by (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938, see example. - Philippe Deléham, Feb 13 2013
Also the Bell transform of A000244. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			[1]; [3,1]; [9,9,1]; ...; p(3,x) = x*(9 + 9*x + x^2).
From _Philippe Deléham_, Feb 13 2013: (Start)
Triangle (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
  1;
  0,   1;
  0,   3,   1;
  0,   9,   9,   1;
  0,  27,  63,  18,   1;
  0,  81, 405, 225,  30,   1;
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000244, A016137, A017933, A028085, A075515, A075516, A075906. Row sums are A004212.

Cf. A075497, A075499.

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> 3^n, 9); # Peter Luschny, Jan 26 2016

Flatten[Table[3^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
rows = 9;
t = Table[3^n, {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

for(n=1, 11, for(m=1, n, print1(3^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (3^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*3)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 3*m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-3*k*x), m >= 1.
E.g.f. for m-th column: (((exp(3*x)-1)/3)^m)/m!, m >= 1.
From Peter Bala, Jan 13 2018: (Start)
Dobinski-type formulas for row polynomials R(n,x):
R(n,x) = exp(-x/3)*Sum_{i >= 0} (3*i)^n* (x/3)^i/i!;
R(n+1,x) = x*exp(-x/3)*Sum_{i >= 0} (3 + 3*i)^n* (x/3)^i/i!.
R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*3^(n-k)*R(k,x).(End)

A016152 a(n) = 4^(n-1)*(2^n-1).

Original entry on oeis.org

0, 1, 12, 112, 960, 7936, 64512, 520192, 4177920, 33488896, 268173312, 2146435072, 17175674880, 137422176256, 1099444518912, 8795824586752, 70367670435840, 562945658454016, 4503582447501312, 36028728299487232

Offset: 0

N. J. A. Sloane

Numbers whose binary representation is the concatenation of n digits 1 and 2(n-1) digits 0, for n>0. (See A147816.) - Omar E. Pol, Nov 13 2008
a(n) is the number of lattices L in Z^n such that the quotient group Z^n / L is C_8. - Álvar Ibeas, Nov 29 2015
a(n) is a maximum number of intercalates in a Latin square of order 2^n (see A092237). - Eduard I. Vatutin, Apr 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..140
Eduard I. Vatutin, Example of Latin squares of order 2^n with maximum number of intercalates.
Index entries for linear recurrences with constant coefficients, signature (12,-32).

Second column of triangle A075499.
Cf. A019677, A147538, A147816.
Cf. A065442, A092237.

[4^(n-1)*(2^n-1): n in [0..40]]; // Vincenzo Librandi, Apr 26 2011

Table[4^(n - 1) (2^n - 1), {n, 0, 19}] (* Michael De Vlieger, Nov 30 2015 *)

a(n)=4^(n-1)*(2^n-1) \\ Charles R Greathouse IV, Oct 07 2015

my(x='x+O('x^30)); concat(0, Vec(x/((1-4*x)*(1-8*x)))) \\ Altug Alkan, Dec 04 2015

[lucas_number1(n,12,32) for n in range(0, 20)] # Zerinvary Lajos, Apr 27 2009

From Barry E. Williams, Jan 17 2000: (Start)
a(n) = ((8^(n+1)) - 4^(n+1))/4.
a(n) = 12a(n-1) - 32a(n-2), n>0; a(0)=1. (End)
a(n) = (4^(n-1))*Stirling2(n+1, 2), n>=0, with Stirling2(n, m)=A008277(n, m).
a(n) = -4^(n-1) + 2*8^(n-1).
E.g.f. for a(n+1), n>=0: d^2/dx^2((((exp(4*x)-1)/4)^2)/2!) = -exp(4*x) + 2*exp(8*x).
G.f.: x/((1-4*x)*(1-8*x)).
((6+sqrt4)^n - (6-sqrt4)^n)/4 in Fibonacci form. Offset 1. a(3)=112. - Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008
a(n) + A160873(n) + A006096(n) = A006096(n+2), for n > 2. - Álvar Ibeas, Nov 29 2015
Sum_{n>0} 1/a(n) = 4*E - 16/3, where E is the Erdős-Borwein constant (A065442). - Peter McNair, Dec 19 2022

A075500 Stirling2 triangle with scaled diagonals (powers of 5).

Original entry on oeis.org

1, 5, 1, 25, 15, 1, 125, 175, 30, 1, 625, 1875, 625, 50, 1, 3125, 19375, 11250, 1625, 75, 1, 15625, 196875, 188125, 43750, 3500, 105, 1, 78125, 1984375, 3018750, 1063125, 131250, 6650, 140, 1, 390625, 19921875

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.

The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(5*z) - 1)*x/5) - 1.

			[1]; [5,1]; [25,15,1]; ...; p(3,x) = x(25 + 15*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*     1
*     5       1
*    25      15       1
*   125     175      30       1
*   625    1875     625      50      1
*  3125   19375   11250    1625     75    1
* 15625  196875  188125   43750   3500  105   1
* 78125 1984375 3018750 1063125 131250 6650 140 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000351, A016164, A075911-A075915. Row sums are A005011(n-1).

Cf. A075499, A075501.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 5^n, 9); # Peter Luschny, Jan 28 2016

Flatten[Table[5^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[5^#&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

for(n=1, 11, for(m=1, n, print1(5^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (5^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*5)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 5m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-5k*x), m >= 1.
E.g.f. for m-th column: (((exp(5x)-1)/5)^m)/m!, m >= 1.

A111578 Triangle T(n, m) = T(n-1, m-1) + (4m-3)*T(n-1, m) read by rows 1<=m<=n.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 31, 15, 1, 1, 156, 166, 28, 1, 1, 781, 1650, 530, 45, 1, 1, 3906, 15631, 8540, 1295, 66, 1, 1, 19531, 144585, 126651, 30555, 2681, 91, 1, 1, 97656, 1320796, 1791048, 646086, 86856, 4956, 120, 1, 1, 488281, 11984820, 24604420, 12774510

Offset: 1

Gary W. Adamson, Aug 07 2005

From Peter Bala, Jan 27 2015: (Start)
Working with an offset of 0, this is the exponential Riordan array [exp(z), (exp(4*z) - 1)/4].
This is the triangle of connection constants between the polynomial basis sequences {x^n}n>=0 and { n!*4^n * binomial((x - 1)/4,n) }n>=0. An example is given below.
Call this array M and let P denote Pascal's triangle A007318 then P^2 * M = A225469; P^(-1) * M is a shifted version of A075499.
This triangle is the particular case a = 4, b = 0, c = 1 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. (End)

			The triangle starts in row n=1 as:
  1;
  1,1;
  1,6,1;
  1,31,15,1;
  1,156,166,28,1;
Connection constants: Row 4: [1, 31, 15, 1] so
x^3 = 1 + 31*(x - 1) + 15*(x - 1)*(x - 5) + (x - 1)*(x - 5)*(x - 9). - _Peter Bala_, Jan 27 2015

P. Bala, A 3 parameter family of generalized Stirling numbers

Cf. A111577, A008277, A039755, A016234 (3rd column).

T[n_, k_] := 1/(4^(k-1)*(k-1)!) * Sum[ (-1)^(k-j-1) * (4*j+1)^(n-1) * Binomial[k-1, j], {j, 0, k-1}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Peter Bala *)

def A096038(n,m):
    if n < 1 or m < 1 or m > n:
        return 0
    elif n <=2:
        return 1
    else:
        return A096038(n-1,m-1)+(4*m-3)*A096038(n-1,m)
print( [A096038(n,m) for n in range(20) for m in range(1,n+1)] )
# R. J. Mathar, Oct 11 2009

From Peter Bala, Jan 27 2015: (Start)
The following formulas assume an offset of 0.
T(n,k) = 1/(4^k*k!)*sum {j = 0..k} (-1)^(k-j)*binomial(k,j)*(4*j + 1)^n.
T(n,k) = sum {i = 0..n-1} 4^(i-k+1)*binomial(n-1,i)*Stirling2(i,k-1).
E.g.f.: exp(z)*exp(x/4*(exp(4*z) - 1)) = 1 + (1 + x)*z + (1 + 6*x + x^2)*z^2/2! + ....
O.g.f. for n-th diagonal: exp(-x/4)*sum {k >= 0} (4*k + 1)^(k + n - 1)*((x/4*exp(-x))^k)/k!.
O.g.f. column k: 1/( (1 - x)*(1 - 5*x)*...*(1 - (4*k + 1)*x) ). (End)

Edited and extended by R. J. Mathar, Oct 11 2009

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A075907 Fourth column of triangle A075499.

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A075908 Fifth column of triangle A075499.

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A075910 Seventh column of triangle A075499.

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A075909 Sixth column of triangle A075499.

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A099394 Triangle T(k,n) by rows: n! * A075499(k,n).

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A004213 Shifts one place left under 4th-order binomial transform.

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A075498 Stirling2 triangle with scaled diagonals (powers of 3).

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A016152 a(n) = 4^(n-1)*(2^n-1).

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