A007318 -id:A007318 - cp's OEIS frontend

A135225 Pascal's triangle A007318 augmented with a leftmost border column of 1's.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 4, 1, 1, 1, 5, 10, 10, 5, 1, 1, 1, 6, 15, 20, 15, 6, 1, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums give A094373.
From Peter Bala, Sep 08 2011: (Start)
This augmented Pascal array, call it P, has interesting connections with the Bernoulli polynomials B(n,x). The infinitesimal generator S of P is the array such that exp(S) = P. The array S is obtained by augmenting the infinitesimal generator A132440 of the Pascal triangle with an initial column [0, 0, 1/2, 1/6, 0, -1/30, ...] on the left. The entries in this column, after the first two zeros, are the Bernoulli values B(n,1), n>=1.
The array P is also connected with the problem of summing powers of consecutive integers. In the array P^n, the entry in position p+1 of the first column is equal to sum {k = 1..n} k^p - see the Example section below.
For similar results for the square of Pascal's triangle see A062715.
Note: If we augment Pascal's triangle with the column [1, 1, x, x^2, x^3, ...] on the left, the resulting lower unit triangular array has the Bernoulli polynomials B(n,x) in the first column of its infinitesimal generator. The present case is when x = 1.
(End)

			First few rows of the triangle:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 2, 1;
  1, 1, 3, 3, 1;
  1, 1, 4, 6, 4, 1;
...
The infinitesimal generator for P begins:
  /0
  |0.......0
  |1/2.....1...0
  |1/6.....0...2....0
  |0.......0...0....3....0
  |-1/30...0...0....0....4....0
  |0.......0...0....0....0....5....0
  |1/42....0...0....0....0....0....6....0
  |...
  \
The array P^n begins:
  /1
  |1+1+...+1........1
  |1+2+...+n........n.........1
  |1+2^2+...+n^2....n^2.....2*n........1
  |1+2^3+...+n^3....n^3.....3*n^2....3*n.......1
  |...
  \
More generally, the array P^t, defined as exp(t*S) for complex t, begins:
  /1
  |B(1,1+t)-B(1,1)..........1
  |1/2*(B(2,1+t)-B(2,1))....t.........1
  |1/3*(B(3,1+t)-B(3,1))....t^2.....2*t........1
  |1/4*(B(4,1+t)-B(4,1))....t^3.....3*t^2....3*t.......1
  |...
  \

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A007318, A027641, A027642, A062715, A094373, A103438, A132440.

T:= function(n,k)
    if k=0 then return 1;
    else return Binomial(n-1,k-1);
    fi; end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 19 2019

T:= func< n, k | k eq 0 select 1 else Binomial(n-1, k-1) >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 19 2019

T:= proc(n, k) option remember;
      if k=0 then 1
    else binomial(n-1, k-1)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 19 2019

T[n_, k_]:= T[n, k]= If[k==0, 1, Binomial[n-1, k-1]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 19 2019 *)

T(n,k) = if(k==0, 1, binomial(n-1, k-1)); \\ G. C. Greubel, Nov 19 2019

@CachedFunction
def T(n,k):
    if (k==0): return 1
    else: return binomial(n-1, k-1)
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 19 2019

A103451 * A007318 * A000012(signed), where A000012(signed) = (1; -1,1; 1,-1,1; ...); as infinite lower triangular matrices.

Given A007318, binomial(n,k) is shifted to T(n+1,k+1) and a leftmost border of 1's is added.

Corrected by R. J. Mathar, Apr 16 2013

A152828 Triangle read by rows, A007318 rows repeated three times .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 6, 15, 20, 15, 6, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 7, 21

Offset: 0

Philippe Deléham, Dec 13 2008

Diagonal sums : A079398 . Lengths of row are : 1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,... (A008620) .

			Triangle begins : 1 ; 1 ; 1 ; 1,1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; 1,3,3,1 ; 1,3,3,1 ; 1,4,6,4,1 ; ...

Cf. A007318, A152198, A152815

{#,#,#}&/@Table[Binomial[n,k],{n,0,11},{k,0,n}]//Flatten (* Harvey P. Dale, Jul 22 2024 *)

A152830 Triangle read by rows, A007318 rows repeated four times .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 6, 15, 20, 15, 6, 1, 1

Offset: 0

Philippe Deléham, Dec 13 2008

Diagonal sums : A103372 .

			Triangle begins : 1 ; 1 ; 1 ; 1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A007318, A152198, A152815, A152828.

Flatten[Table[#,{4}]&/@Table[Binomial[n,k],{n,0,6},{k,0,n}]] (* Harvey P. Dale, Sep 23 2015 *)

A059110 Triangle T = A007318A271703; T(n,m)= Sum_{i=0..n} L'(n,i)binomial(i,m), m=0..n.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 13, 21, 9, 1, 73, 136, 78, 16, 1, 501, 1045, 730, 210, 25, 1, 4051, 9276, 7515, 2720, 465, 36, 1, 37633, 93289, 85071, 36575, 8015, 903, 49, 1, 394353, 1047376, 1053724, 519456, 137270, 20048, 1596, 64, 1, 4596553, 12975561

Offset: 0

Vladeta Jovovic, Jan 04 2001

L'(n,i) are unsigned Lah numbers (cf. A008297): L'(n,i)=n!/i!*binomial(n-1,i-1) for i >= 1, L'(0,0)=1, L'(n,0)=0 for n>0. T(n,0)=A000262(n); T(n,2)=A052852(n). Row sums A052897.
Exponential Riordan array [e^(x/(1-x)),x/(1-x)]. - Paul Barry, Apr 28 2007
From Wolfdieter Lang, Jun 22 2017: (Start)
The inverse matrix T^(-1) is exponential Riordan (aka Sheffer) (e^(-x), x/(1+x)): T^(-1)(n, m) = (-1)^(n-m)*A271705(n, m).
The a- and z-sequences of this Sheffer (aka exponential Riordan) matrix are a = [1,1,repeat(0)] and z(n) = (-1)^(n+1)*A028310(n)/A000027(n-1) with e.g.f. ((1+x)/x)*(1-exp(-x)). For a- and z-sequences see a W. Lang link under A006232 with references. (End)

			The triangle T = A007318*A271703 starts:
n\m       0        1        2       3       4      5     6    7  8 9 ...
0:        1
1:        1        1
2:        3        4        1
3:       13       21        9       1
4:       73      136       78      16       1
5:      501     1045      730     210      25      1
6:     4051     9276     7515    2720     465     36     1
7:    37633    93289    85071   36575    8015    903    49    1
8:   394353  1047376  1053724  519456  137270  20048  1596   64  1
9:  4596553 12975561 14196708 7836276 2404206 427518 44436 2628 81 1
... reformatted. - _Wolfdieter Lang_, Jun 22 2017
E.g.f. for T(n, 2) = 1/2!*(x/(1-x))^2*e^(x/(x-1)) = 1*x^2/2 + 9*x^3/3! + 78*x^4/4! + 730*x^5/5! + 7515*x^6/6 + ...
From _Wolfdieter Lang_, Jun 22 2017: (Start)
The z-sequence starts: [1, 1/2, -2/3, 3/4, -4/5, 5/6, -6/7, 7/8, -8/9, ...
T recurrence: T(3, 0) = 3*(1*T(2,0) + (1/2)*T(2, 1) + (-2/3)*T(2 ,1)) = 3*(3 + (1/2)*4 - (2/3)) = 13; T(3, 1) = 3*(T(2, 0)/1 + T(2, 1)) = 3*(3 + 4) = 21.
Meixner type recurrence for R(2, x): (D - D^2)*(3 + 4*x + x^2) = 4 + 2*x - 2 = 2*(1 + x), (D = d/dx).
General Sheffer recurrence for R(2, x): (1+x)*(1 + 2*D + D^2)*(1 + x) = (1+x)*(1 + x + 2) = 3 + 4*x + x^2. (End)

Muniru A Asiru, Rows n=0..50 of triangle, flattened
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Index entries for sequences related to Laguerre polynomials

Cf. A000262, A007318, A008297, A052852, A052897, A271703, A271705.

Concatenation([1],Flat(List([1..10],n->List([0..n],m->Sum([0..n],i-> Factorial(n)/Factorial(i)*Binomial(n-1,i-1)*Binomial(i,m)))))); # Muniru A Asiru, Jul 25 2018

A059110:= func< n,k | n eq 0 select 1 else Factorial(n-1)*Binomial(n,k)*Evaluate(LaguerrePolynomial(n-1, 1-k), -1) >;
[A059110(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 23 2021

Lprime := proc(n,i)
    if n = 0 and i = 0 then
        1;
    elif k = 0 then
        0 ;
    else
        n!/i!*binomial(n-1,i-1) ;
    end if;
end proc:
A059110 := proc(n,k)
    add(Lprime(n,i)*binomial(i,k),i=0..n) ;
end proc: # R. J. Mathar, Mar 15 2013

(* First program *)
lp[n_, i_] := Binomial[n-1, i-1]*n!/i!; lp[0, 0] = 1; t[n_, m_] := Sum[lp[n, i]*Binomial[i, m], {i, 0, n}]; Table[t[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* Jean-François Alcover, Mar 26 2013 *)
(* Second program *)
A059110[n_, k_]:= If[n==0, 1, (n-1)!*Binomial[n, k]*LaguerreL[n-1, 1-k, -1]];
Table[A059110[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 23 2021 *)

def A059110(n, k): return 1 if n==0 else factorial(n-1)*binomial(n, k)*gen_laguerre(n-1, 1-k, -1)
flatten([[A059110(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 23 2021

E.g.f. for column m: (1/m!)*(x/(1-x))^m*e^(x/(x-1)), m >= 0.
From Wolfdieter Lang, Jun 22 2017: (Start)
E.g.f. for row polynomials in powers of x (e.g.f. of the triangle): exp(z/(1-z))* exp(x*z/(1-z)) (exponential Riordan).
Recurrence: T(n, 0) = Sum_{j=0} z(j)*T(n-1, j), n >= 1, with z(n) = (-1)^(n+1)*A028310(n), T(0, 0) = 1, T(n, m) = 0 n < m, T(n, m) = n*(T(n-1, m-1)/m + T(n-1, m)), n >= m >= 1 (from the z- and a-sequence, see a comment above).
Meixner type recurrence for the (monic) row polynomials R(n, x) = Sum_{m=0..n} T(n, m)*x^m: Sum_{k=0..n-1} (-1)^k*D^(k+1)*R(n, x) = n*R(n-1, x), n >=1, R(0, x) = 1, with D = d/dx.
General Sheffer recurrence: R(n, x) = (x+1)*(1+D)^2*R(n-1, x), n >=1, R(0, x) = 1.
(End)
P_n(x) = L_n(1+x) = n!*Lag_n(-(1+x);1), where P_n(x) are the row polynomials of this entry; L_n(x), the Lah polynomials of A105278; and Lag_n(x;1), the Laguerre polynomials of order 1. These relations follow from the relation between the iterated operator (x^2 D)^n and ((1+x)^2 D)^n with D = d/dx. - Tom Copeland, Jul 18 2018
From G. C. Greubel, Feb 23 2021: (Start)
T(n, k) = (n-1)!*binomial(n, k)*LaguerreL(n-1, 1-k, -1) with T(0, 0) = 1.
Sum_{k=0..n} T(n, k) = A052897(n). (End)

A152831 Triangle read by rows, A007318 repeated five times .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5

Offset: 0

Philippe Deléham, Dec 13 2008

Diagonal sums : A103373 .

			Triangle begins : 1 ; 1 ; 1 ; 1 ; 1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A007318, A152198, A152815, A152828, A152830.

Flatten[Table[#, {5}] & /@ Table[Binomial[n, k], {n,0,10}, {k, 0, n}]] (* G. C. Greubel, May 03 2017 *)

A168287 T(n,k) = 2*A046802(n+1,k+1) - A007318(n,k), triangle read by rows (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 26, 60, 26, 1, 1, 57, 252, 252, 57, 1, 1, 120, 931, 1746, 931, 120, 1, 1, 247, 3201, 10187, 10187, 3201, 247, 1, 1, 502, 10534, 53542, 89788, 53542, 10534, 502, 1, 1, 1013, 33698, 262466, 688976, 688976, 262466, 33698, 1013

Offset: 0

Roger L. Bagula, Nov 22 2009

			Triangle begins:
     1;
     1,    1;
     1,    4,     1;
     1,   11,    11,     1;
     1,   26,    60,    26,     1;
     1,   57,   252,   252,    57,     1;
     1,  120,   931,  1746,   931,   120,     1;
     1,  247,  3201, 10187, 10187,  3201,   247,   1;
     1,  502, 10534, 53542, 89788, 53542, 10534, 502, 1;
     ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A142175, A168288, A168289, A168290, A168291, A168292, A168293.

p[t_] = 2*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - Exp[t*(1 + x)];
Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten

A046802(n, k) := sum(binomial(n - 1, r)*sum(j!*(-1)^(k - j - 1)*stirling2(r, j)*binomial(r - j, k - j - 1), j, 0, k - 1), r, k - 1, n - 1)$
T(n, k) := 2*A046802(n + 1, k + 1) - binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

E.g.f.: 2*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - exp(t*(1 + x)).

Edited, and new name by Franck Maminirina Ramaharo, Oct 21 2018

A168288 T(n,k) = 3A046802(n+1,k+1) - 2A007318(n,k), triangle read by rows (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 37, 87, 37, 1, 1, 83, 373, 373, 83, 1, 1, 177, 1389, 2609, 1389, 177, 1, 1, 367, 4791, 15263, 15263, 4791, 367, 1, 1, 749, 15787, 80285, 134647, 80285, 15787, 749, 1, 1, 1515, 50529, 393657, 1033401, 1033401, 393657, 50529

Offset: 0

Roger L. Bagula, Nov 22 2009

			Triangle begins:
     1;
     1,   1;
     1,   5,     1;
     1,  15,    15,     1;
     1,  37,    87,    37,      1;
     1,  83,   373,   373,     83,     1;
     1, 177,  1389,  2609,   1389,   177,     1;
     1, 367,  4791, 15263,  15263,  4791,   367,   1;
     1, 749, 15787, 80285, 134647, 80285, 15787, 749, 1;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A142175, A168287, A168289, A168290, A168291, A168292, A168293.

p[t_] = 3*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 2*Exp[t*(1 + x)];
Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten

A046802(n, k) := sum(binomial(n - 1, r)*sum(j!*(-1)^(k - j - 1)*stirling2(r, j)*binomial(r - j, k - j - 1), j, 0, k - 1), r, k - 1, n - 1)$
T(n, k) := 3*A046802(n + 1, k + 1) - 2*binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

E.g.f.: 3*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 2*exp(t*(1 + x)).

Edited, and new name by Franck Maminirina Ramaharo, Oct 21 2018

A168289 T(n,k) = 4A046802(n+1,k+1) - 3A007318(n,k), triangle read by rows (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 19, 19, 1, 1, 48, 114, 48, 1, 1, 109, 494, 494, 109, 1, 1, 234, 1847, 3472, 1847, 234, 1, 1, 487, 6381, 20339, 20339, 6381, 487, 1, 1, 996, 21040, 107028, 179506, 107028, 21040, 996, 1, 1, 2017, 67360, 524848, 1377826, 1377826, 524848

Offset: 0

Roger L. Bagula, Nov 22 2009

			Triangle begins:
     1;
     1,   1;
     1,   6,     1;
     1,  19,    19,      1;
     1,  48,   114,     48,      1;
     1, 109,   494,    494,    109,      1;
     1, 234,  1847,   3472,   1847,    234,     1;
     1, 487,  6381,  20339,  20339,   6381,   487,   1;
     1, 996, 21040, 107028, 179506, 107028, 21040, 996, 1;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A142175, A168287, A168288, A168290, A168291, A168292, A168293.

p[t_] = 4*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 3*Exp[t*(1 + x)];
Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten

A046802(n, k) := sum(binomial(n - 1, r)*sum(j!*(-1)^(k - j - 1)*stirling2(r, j)*binomial(r - j, k - j - 1), j, 0, k - 1), r, k - 1, n - 1)$
T(n, k) := 4*A046802(n + 1, k + 1) - 3*binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

E.g.f: 4*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 3*exp(t*(1 + x)).

Edited, and new name by Franck Maminirina Ramaharo, Oct 21 2018

A168290 T(n,k) = 5A046802(n+1,k+1) - 4A007318(n,k), triangle read by rows (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 23, 23, 1, 1, 59, 141, 59, 1, 1, 135, 615, 615, 135, 1, 1, 291, 2305, 4335, 2305, 291, 1, 1, 607, 7971, 25415, 25415, 7971, 607, 1, 1, 1243, 26293, 133771, 224365, 133771, 26293, 1243, 1, 1, 2519, 84191, 656039, 1722251, 1722251, 656039

Offset: 0

Roger L. Bagula, Nov 22 2009

			Triangle begins:
    1;
    1,    1;
    1,    7,     1;
    1,   23,    23,      1;
    1,   59,   141,     59,      1;
    1,  135,   615,    615,    135,      1;
    1,  291,  2305,   4335,   2305,    291,     1;
    1,  607,  7971,  25415,  25415,   7971,   607,    1;
    1, 1243, 26293, 133771, 224365, 133771, 26293, 1243, 1;
     ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A142175, A168287, A168288, A168289, A168291, A168292, A168293.

p[t_] = 5*(1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)]) - 4*Exp[t*(1 + x)];
Table[CoefficientList[FullSimplify[n!*SeriesCoefficient[Series[p[ t], {t, 0, n}], n]], x], {n, 0, 10}]//Flatten

A046802(n, k) := sum(binomial(n - 1, r)*sum(j!*(-1)^(k - j - 1)*stirling2(r, j)*binomial(r - j, k - j - 1), j, 0, k - 1), r, k - 1, n - 1)$
T(n, k) := 5*A046802(n + 1, k + 1) - 4*binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

E.g.f.: 5*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 4*exp(t*(1 + x)).

Edited, new name by Franck Maminirina Ramaharo, Oct 21 2018

A168291 T(n,k) = 4A046802(n+1,k+1) - 2A008518(n,k) - A007318(n,k), triangle read by rows (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 32, 82, 32, 1, 1, 65, 330, 330, 65, 1, 1, 130, 1159, 2304, 1159, 130, 1, 1, 259, 3801, 13195, 13195, 3801, 259, 1, 1, 516, 12016, 67316, 117170, 67316, 12016, 516, 1, 1, 1029, 37212, 319332, 889230, 889230, 319332, 37212

Offset: 0

Roger L. Bagula, Nov 22 2009

			Triangle begins:
     1;
     1,   1;
     1,   6,     1;
     1,  15,    15,     1;
     1,  32,    82,    32,      1;
     1,  65,   330,   330,     65,     1;
     1, 130,  1159,  2304,   1159,   130,     1;
     1, 259,  3801, 13195,  13195,  3801,   259,   1;
     1, 516, 12016, 67316, 117170, 67316, 12016, 516, 1;
      ... reformatted. - _Franck Maminirina Ramaharo_, Oct 21 2018

Triangles related to Eulerian numbers: A008292, A046802, A060187, A123125.

Cf. A142147, A142175, A168287, A168288, A168289, A168290, A168292, A168293.

A123125(n, k) := sum((-1)^(k - j)*(binomial(n - j, k - j))*stirling2(n, j)*j!, j, 0, k)$
A046802(n, k) := sum(binomial(n - 1, r)*A123125(r, k - 1), r, k - 1, n - 1)$
A008518(n, k) := A123125(n, k) + A123125(n, k + 1)$
T(n, k) := 4*A046802(n + 1, k + 1) - 2*A008518(n, k) - binomial(n, k)$
create_list(T(n, k), n, 0, 10, k, 0, n);
/* Franck Maminirina Ramaharo, Oct 21 2018 */

E.g.f.: 4*(1 - x)*exp(t)/(1 - x*exp(t*(1 - x))) - 2*(exp(t) - x*exp(t*x))/(exp(t*x) - x*exp(t)) - exp(t*(1 + x)).

Edited, new name by Franck Maminirina Ramaharo, Oct 21 2018

cp's OEIS Frontend

A135225 Pascal's triangle A007318 augmented with a leftmost border column of 1's.

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A152828 Triangle read by rows, A007318 rows repeated three times .

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A152830 Triangle read by rows, A007318 rows repeated four times .

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A059110 Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.

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A152831 Triangle read by rows, A007318 repeated five times .

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A168287 T(n,k) = 2*A046802(n+1,k+1) - A007318(n,k), triangle read by rows (0 <= k <= n).

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Maxima

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A168288 T(n,k) = 3*A046802(n+1,k+1) - 2*A007318(n,k), triangle read by rows (0 <= k <= n).

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A059110 Triangle T = A007318A271703; T(n,m)= Sum_{i=0..n} L'(n,i)binomial(i,m), m=0..n.

A168288 T(n,k) = 3A046802(n+1,k+1) - 2A007318(n,k), triangle read by rows (0 <= k <= n).

A168289 T(n,k) = 4A046802(n+1,k+1) - 3A007318(n,k), triangle read by rows (0 <= k <= n).

A168290 T(n,k) = 5A046802(n+1,k+1) - 4A007318(n,k), triangle read by rows (0 <= k <= n).

A168291 T(n,k) = 4A046802(n+1,k+1) - 2A008518(n,k) - A007318(n,k), triangle read by rows (0 <= k <= n).