A008949 -id:A008949 - cp's OEIS frontend

1, 3, 1, 8, 4, 1, 20, 12, 5, 1, 48, 32, 17, 6, 1, 112, 80, 49, 23, 7, 1, 256, 192, 129, 72, 30, 8, 1, 576, 448, 321, 201, 102, 38, 9, 1, 1280, 1024, 769, 522, 303, 140, 47, 10, 1, 2816, 2304, 1793, 1291, 825, 443, 187, 57, 11, 1, 6144, 5120, 4097, 3084, 2116, 1268, 630

Offset: 0

Wolfdieter Lang, May 26 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is ((1-z)/(1-2*z)^2)/(1-x*z/(1-z)).
This is the second member of the family of Riordan-type matrices obtained from A007318(n,m) (Pascal's triangle read as lower triangular matrix) by repeated application of the prs-procedure.
The column sequences appear in A001792, A001787, A000337, A045618, A045889, A034009, A055250, A055251 for m=0..7.

			1;
3,1;
8,4,1;
20,12,5,1;
...
Fourth row polynomial (n=3): p(3,x)= 20+12*x+5*x^2+x^3

G. C. Greubel, Table of n, a(n) for n = 0..1274

Cf. A007318, A055248, A008949. Row sums: A049611(n+1) = A055252(n, 0).

a[n_, m_] := Binomial[n, m]*Hypergeometric2F1[2, m-n, m+1, -1]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Mar 11 2014 *)

a(n, m) = Sum_{k=m,..,n} ( A055248(n, k) ), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m) = Sum_{j=m,..,(n-1)} ( a(j, m) ) + A055248(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: ((1-x)/(1-2*x)^2)*(x/(1-x))^m, m >= 0.

a(n, m) = binomial(n, m) * 2F1(2, m-n; m+1; -1) where 2F1 is the hypergeometric function. Jean-François Alcover, Mar 11 2014

A052553 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 0, 0, 1, 5, 6, 1, 0, 0, 0, 1, 6, 10, 4, 0, 0, 0, 0, 1, 7, 15, 10, 1, 0, 0, 0, 0, 1, 8, 21, 20, 5, 0, 0, 0, 0, 0, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 0, 1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0, 0, 1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0, 0, 1, 12, 55

Offset: 0

N. J. A. Sloane, Mar 17 2000

Another version of Pascal's triangle A007318.
As a triangle read by rows, it is (1,0,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938 and it is the Riordan array (1/(1-x), x^2/(1-x)). The row sums of this triangle are F(n+1) = A000045(n+1). - Philippe Deléham, Dec 11 2011
As a triangle, binomial(n-k, k) is also the number of ways to add k pierced circles to a path graph P_n so that no two circles share a vertex (see Lemma 3.1 at page 5 in Owad and Tsvietkova). - Stefano Spezia, May 18 2022
For all n >= 0, k >= 0, the k-th homology group of the n-torus H_k(T^n) is the free abelian group of rank T(n,k) = binomial(n,k). See the Math Stack Exchange link below. - Jianing Song, Mar 13 2023

			Array begins:
  1, 0,  0,  0, 0, 0, ...
  1, 1,  0,  0, 0, 0, ...
  1, 2,  1,  0, 0, 0, ...
  1, 3,  3,  1, 0, 0, ...
  1, 4,  6,  4, 1, 0, ...
  1, 5, 10, 10, 5, 1, ...
As a triangle, this begins:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  0, 0;
  1, 3,  1, 0, 0;
  1, 4,  3, 0, 0, 0;
  1, 5,  6, 1, 0, 0, 0;
  1, 6, 10, 4, 0, 0, 0, 0;
  ...

Vincenzo Librandi, Table of n, a(n) for n = 0..5459
Math Stack Exchange, Homology of the n-torus using the Künneth Formula
Nicholas Owad and Anastasiia Tsvietkova, Random meander model for links, arXiv:2205.03451 [math.GT], 2022.
Index entries for triangles and arrays related to Pascal's triangle

The official entry for Pascal's triangle is A007318. See also A026729 (the same array read by downward antidiagonals).
As a triangle without zeros: A011973.
Cf. A052509, A054123, A054124, A008949.

/* As triangle */ [[Binomial(n-k,k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 08 2017

with(combinat): for s from 0 to 20 do for n from s to 0 by -1 do printf(`%d,`, binomial(n, s-n)) od:od: # James Sellers, Mar 17 2000

Flatten[ Table[ Binomial[n-k , k], {n, 0, 13}, {k, 0, n}]]  (* Jean-François Alcover, Dec 05 2012 *)

T(n,k) = binomial(n,k) \\ Charles R Greathouse IV, Feb 07 2017

As a triangle: T(n,k) = A026729(n,n-k).

G.f. of the triangular version: 1/(1-x-x^2*y). - R. J. Mathar, Aug 11 2015

A102661 Triangle of partial sums of Stirling numbers of 2nd kind (A008277): T(n,k) = Sum_{i=1..k} Stirling2(n,i), 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 1, 8, 14, 15, 1, 16, 41, 51, 52, 1, 32, 122, 187, 202, 203, 1, 64, 365, 715, 855, 876, 877, 1, 128, 1094, 2795, 3845, 4111, 4139, 4140, 1, 256, 3281, 11051, 18002, 20648, 21110, 21146, 21147, 1, 512, 9842, 43947, 86472, 109299, 115179, 115929, 115974, 115975

Offset: 1

Vladeta Jovovic, Feb 03 2005

T(n,k) is the number of ways to place n distinguishable balls into k indistinguishable bins. - Geoffrey Critzer, Mar 22 2011
From Mark Wildon, Aug 10 2015: (Start)
T(n,k) is the number of partitions of a set of size n into at most k parts.
T(n,k) is the number of sequences of n top-to-random shuffles of a deck of k cards that leave the deck invariant.
T(n,k) =  where pi is the natural permutation character of the symmetric group Sym_k. This gives another combinatorial interpretation of T(n,k) as counting sequences of box moves on Young diagrams. Reference linked to below. (End)
Diagonal entries T(n,n) are the Bell numbers A000110. - Robert Israel, Aug 10 2015
From Manfred Boergens, Mar 18 2025: (Start)
The partitions in the second comment can be described as disjoint collections of subsets of [n] without the empty set with union = [n]. For instance, T(4,2)=8 is the number of partitions of [4] into 1 or 2 parts: 1234, 1 234, 2 134, 3 124, 4 123, 12 34, 13 24, 14 23.
For disjoint collections which may include one empty set see A381682.
For arbitrary collections without the empty set see A369950.
For arbitrary collections which may include one empty set see A381683. (End)

			Triangle begins:
  1;
  1,  2;
  1,  4,  5;
  1,  8, 14, 15;
  1, 16, 41, 51, 52;
  ...

Richard Stanley, Enumerative Combinatorics, Cambridge Univ. Press, 1997 page 38. (#7 of the twelvefold ways)

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
John R. Britnell and Mark Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D, arXiv:1507.04803 [math.CO], 2015.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers

Cf. A000110, A008277, A008949, A048993, A049444.

Cf. A369950, A381682, A381683.

a102661 n k = a102661_tabl !! (n-1) !! (k-1)
a102661_row n = a102661_tabl !! (n-1)
a102661_tabl = map (scanl1 (+) . tail) $ tail a048993_tabl
-- Reinhard Zumkeller, Jun 19 2015

with(combinat): A102661_row := proc(n) local k,j; seq(add(stirling2(n,j),j=1..k),k=1..n) end:
seq(print(A102661_row(r)),r=1..6); # Peter Luschny, Sep 30 2011

Table[Table[Sum[StirlingS2[n, i], {i, 1, k}], {k, 1, n}], {n, 1,10}] // Grid (* Geoffrey Critzer, Mar 22 2011*)
Table[Accumulate[StirlingS2[n,Range[n]]],{n,10}]//Flatten (* Harvey P. Dale, Oct 28 2019 *)

tabl(nn) = {for (n=1, nn, for (k=1, n, print1(sum(i=1, k, stirling(n,i, 2)), ", ");); print(););} \\ Michel Marcus, Aug 10 2015

def T(n,k):
    return sum([stirling_number2(n,j) for j in range(1,k+1)])
# Danny Rorabaugh, Oct 13 2015

E.g.f. for row polynomials s(n,y) = Sum_{k=0..n} a(n,k)*y^k is (y*e^(e^(x*y)-1)- e^(y*(e^x-1)))/(y-1) - 1. - Robert Israel, Aug 10 2015

A090447 Triangle of partial products of binomials.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 9, 9, 1, 4, 24, 96, 96, 1, 5, 50, 500, 2500, 2500, 1, 6, 90, 1800, 27000, 162000, 162000, 1, 7, 147, 5145, 180075, 3781575, 26471025, 26471025, 1, 8, 224, 12544, 878080, 49172480, 1376829440, 11014635520, 11014635520, 1, 9, 324

Offset: 0

Wolfdieter Lang, Dec 23 2003

			[1]; [1,1]; [1,2,2]; [1,3,9,9]; ...

W. Lang, First 10 rows.

Column sequences: A000027 (natural numbers), A006002, A090448-9.
Cf. A090450 (row sums), A090451 (alternating row sums).
Cf. A008949 (partial row sums in Pascal's triangle).
Cf. A000178, A007318, A008279.

a[n_, m_] := Product[Binomial[n, p], {p, 0, m}]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

a(n, m) = Product_{p=0..m} binomial(n, p), n>=m>=0, else 0. Partial row products in Pascal's triangle A007318.
a(n, m) = (Product_{p=0..m} fallfac(n, m-p))/superfac(m), n>=m>=0, else 0; with fallfac(n, m) := A008279(n, m) (falling factorials) and superfac(m) = A000178(m) (superfactorials).
a(n, m) = (Product_{p=0..m} (n-p)^(m-p))/superfac(m), n>=m>=0, with 0^0:=0, else 0.

A261363 Triangle read by rows: partial row sums of Sierpinski's triangle.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 2, 3, 4, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 4, 1, 1, 2, 2, 3, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 1, 2, 3, 4, 4, 4, 4, 4, 5, 6, 7, 8, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4

Offset: 0

Reinhard Zumkeller, Aug 16 2015

T(n,n) = number of distinct terms in row n = number of odd terms in row n+1 = A001316(n).

Central terms, for n > 0: T(2*n,n) = A048896(n-1).

			.   n |  Sierpinski: A047999(n,*)  |  Partial row sums: T(n,*)
. ----+----------------------------+----------------------------
.   0 |              1             |              1
.   1 |             1 1            |             1 2
.   2 |            1 0 1           |            1 1 2
.   3 |           1 1 1 1          |           1 2 3 4
.   4 |          1 0 0 0 1         |          1 1 1 1 2
.   5 |         1 1 0 0 1 1        |         1 2 2 2 3 4
.   6 |        1 0 1 0 1 0 1       |        1 1 2 2 3 3 4
.   7 |       1 1 1 1 1 1 1 1      |       1 2 3 4 5 6 7 8
.   8 |      1 0 0 0 0 0 0 0 1     |      1 1 1 1 1 1 1 1 2
.   9 |     1 1 0 0 0 0 0 0 1 1    |     1 2 2 2 2 2 2 2 3 4
.  10 |    1 0 1 0 0 0 0 0 1 0 1   |    1 1 2 2 2 2 2 2 3 3 4
.  11 |   1 1 1 1 0 0 0 0 1 1 1 1  |   1 2 3 4 4 4 4 4 5 6 7 8
.  12 |  1 0 0 0 1 0 0 0 1 0 0 0 1 |  1 1 1 1 2 2 2 2 3 3 3 3 4  .

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A047999, A008949, A048896 (central terms), A001316 (right edge), A261366.

a261363 n k = a261363_tabl !! n !! k
a261363_row n = a261363_tabl !! n
a261363_tabl = map (scanl1 (+)) a047999_tabl

row[n_] := Accumulate[Array[Boole[0 == BitAnd[n-#, #]] &, n + 1, 0]]; Array[row, 13, 0] // Flatten (* Amiram Eldar, May 13 2025 *)

A054124 Left Fibonacci row-sum array, n >= 0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 4, 1, 1, 1, 2, 4, 7, 5, 1, 1, 1, 2, 4, 8, 11, 6, 1, 1, 1, 2, 4, 8, 15, 16, 7, 1, 1, 1, 2, 4, 8, 16, 26, 22, 8, 1, 1, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 1, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 1, 2

Offset: 0

Clark Kimberling

Reflection of array in A054123 about vertical central line.
Starting with g(0) = {0}, generate g(n) for n > 0 inductively using these rules:
  (1)  if x is in g(n-1), then x+1 is in g(n); and
  (2)  if x is in g(n-1) and x < 2, then x/2 is in g(n).
Then g(1) = {1/1}, g(2) = {1/2,2/1}, g(3) = {1/4,3/2,3/1}, etc. The denominators in g(n) are 2^0, 2^1, ..., 2^(n-1), and T(n,k) is the number of occurrences of 2^k, for k = 0..n-1. - Clark Kimberling, Nov 09 2015
Variant of A004070 with an additional column of 1's on the left. - Jianing Song, May 30 2022

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Row sums: A000045. Central numbers: 1, 1, 2, 4, 8, ... (A000079).

First n numbers of n-th column for n >= 1 form the array in A008949.

a054124 n k = a054124_tabl !! n !! k
a054124_row n = a054124_tabl !! n
a054124_tabl = map reverse a054123_tabl
-- Reinhard Zumkeller, May 26 2015

t[, 0|1] = t[n, n_] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + t[n-2, k-1]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 25 2013 *)

A052509(n,k) = sum(m=0, k, binomial(n-k, m));
T(n,k) = if(k==0, 1, A052509(n-1,n-k)) \\ Jianing Song, May 30 2022

T(n, 0) = T(n, n) = 1 for n >= 0; T(n, 1) = 1 for n >= 1; T(n, k) = T(n-1, k-1) + T(n-2, k-1) for k=2, 3, ..., n-1, n >= 3. [Corrected by Jianing Song, May 30 2022]

G.f.: Sum_{n>=0, 0<=k<=n} T(n,k) * x^n * y^k = (1-x^2*y) / ((1-x)*(1-x*y-x^2*y)). - Jianing Song, May 30 2022

A163866 Partial sums of A007318.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 11, 14, 15, 16, 20, 26, 30, 31, 32, 37, 47, 57, 62, 63, 64, 70, 85, 105, 120, 126, 127, 128, 135, 156, 191, 226, 247, 254, 255, 256, 264, 292, 348, 418, 474, 502, 510, 511, 512, 521, 557, 641, 767, 893, 977, 1013, 1022, 1023, 1024, 1034, 1079, 1199

Offset: 1

Juri-Stepan Gerasimov, Aug 06 2009

nonn

			a(1)=0!/(0!(0-0)!)=1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A008949.

Cf. A259525.

a163866 n = a163866_list !! (n-1)
a163866_list = scanl1 (+) $ concat a007318_tabl
-- Reinhard Zumkeller, Jul 18 2015

Flatten[Table[2^a-1+Sum[Binomial[a,p],{p,0,b}],{a,0,10},{b,0,a}]] (* Frank M Jackson, Apr 25 2011 *)
Accumulate[Flatten[Table[Binomial[n,k],{n,0,11},{k,0,n}]]] (* Harvey P. Dale, Dec 04 2012 *)

lista(nn) = {my(i=0, j=0, p=0); for (n=1, nn, p += binomial(i, j); print1(p, ", "); j++; if (j > i, j = 0; i++););} \\ Michel Marcus, Jan 25 2019

a(n) = Sum_{j=1..n-1} A007318(j).

Entries checked by R. J. Mathar, Aug 11 2009

A249111 Triangle of partial sums of rows in triangle A249095.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 4, 5, 6, 1, 2, 5, 7, 10, 11, 12, 1, 2, 6, 9, 15, 18, 22, 23, 24, 1, 2, 7, 11, 21, 27, 37, 41, 46, 47, 48, 1, 2, 8, 13, 28, 38, 58, 68, 83, 88, 94, 95, 96, 1, 2, 9, 15, 36, 51, 86, 106, 141, 156, 177, 183, 190, 191, 192, 1, 2, 10, 17, 45, 66

Offset: 0

Reinhard Zumkeller, Nov 14 2014

Length of row n = 2*n+1.

In the layout as given in the example, T(n,k) is the sum of the two elements to the left and to the right of the element just above, with the row continued to the left by 0's and to the right by the last element, cf. formula. - M. F. Hasler, Nov 17 2014

			The triangle begins:
.  0:                            1
.  1:                        1   2   3
.  2:                    1   2   4   5   6
.  3:                1   2   5   7  10  11  12
.  4:             1  2   6   9  15  18  22  23  24
.  5:          1  2  7  11  21  27  37  41  46  47  48
.  6:       1  2  8 13  28  38  58  68  83  88  94  95  96
.  7:    1  2  9 15 36  51  86 106 141 156 177 183 190 191 192
.  8:  1 2 10 17 45 66 122 157 227 262 318 339 367 374 382 383 384 .
It can be seen that the elements (except for row 1) are sum of the neighbors to the upper left and upper right, with the table continued to the left with 0's and to the right with the last = largest element of each row. E.g., 1=0+1, 2=0+2, 4=1+3, 5=2+3 (=1+4 in the next row), 6=3+3 (in row 2), 7=2+5 etc. - _M. F. Hasler_, Nov 17 2014

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

Cf. A005408 (row lengths), A128543 (row sums), A248574 (central terms), A008949.

a249111 n k = a249111_tabf !! n !! k
a249111_row n = a249111_tabf !! n
a249111_tabf = map (scanl1 (+)) a249095_tabf

T(n,k)=if(k<2,k+1,if(k>=2*n-2,3<<(n-1),T(n-1,k-2)+T(n-1,k))) \\ M. F. Hasler, Nov 17 2014

T(n,0) = A249095(n,0) = 1; T(n,k) = T(n,k-1) + A249095(n,k), k <= n.

T(n+1,k+1) = T(n,k-1) + T(n,k+1), with T(n,k-1)=0 for k<1 and T(n,k+1)=T(n,k) for k>=2n (last element of the row). In particular, T(n,k)=k+1 if k<2n and T(n,k)=3*2^(n-1) if k>=2n. - M. F. Hasler, Nov 17 2014

A058393 A square array based on 1^n (A000012) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 3, 1, 0, 1, 2, 4, 4, 1, 1, 1, 2, 4, 7, 5, 1, 0, 1, 2, 4, 8, 11, 6, 1, 1, 1, 2, 4, 8, 15, 16, 7, 1, 0, 1, 2, 4, 8, 16, 26, 22, 8, 1, 1, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 0, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 1, 2, 4, 8, 16, 32, 63, 99, 93, 46, 11, 1, 0

Offset: 0

Henry Bottomley, Nov 24 2000

Changing the formula by replacing T(0,2n)=T(1,n) by T(0,2n)=T(m,n) for some other value of m, would make the generating function change to coefficient of x^n in expansion of (1+x)^k/(1-x^2)^m. This would produce A058394, A058395, A057884, (and effectively A007318).

			Rows are (1,0,1,0,1,0,1,...), (1,1,1,1,1,1,...), (1,2,2,2,2,2,...), (1,3,4,4,4,...) etc.

Rows are A000035 (A000012 with zeros), A000012, A040000 etc. Columns are A000012, A001477, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863 etc. Diagonals include A000079, A000225, A000295, A002662, A002663, A002664, A035038, A035039, A035040, A035041, etc. The triangles A008949, A054143 and A055248 also appear in the half of the array which is not powers of 2.

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(1, 1)=1, T(0, 2n)=T(1, n) and T(0, 2n+1)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2).

A193820 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 7, 8, 8, 1, 5, 11, 15, 16, 16, 1, 6, 16, 26, 31, 32, 32, 1, 7, 22, 42, 57, 63, 64, 64, 1, 8, 29, 64, 99, 120, 127, 128, 128, 1, 9, 37, 93, 163, 219, 247, 255, 256, 256, 1, 10, 46, 130, 256, 382, 466, 502, 511, 512, 512, 1, 11, 56

Offset: 0

Clark Kimberling, Aug 06 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Variant of A054143 and A008949. - R. J. Mathar, Mar 03 2013

			First six rows:
  1
  1....1
  1....2....2
  1....3....4....4
  1....4....7....8....8
  1....5....11...15...16...16

Cf. A193722, A128175, A193823, A045623 (row sums), A009766.

Cf. A054143, A008949.

A193820 := (n,k) -> `if`(k=0 or n=0,1, A193820(n-1,k-1)+A193820(n-1,k));
seq(print(seq(A193820(n,k),k=0..n+1)),n=0..10); # Peter Luschny, Jan 22 2012

z = 10; a = 1; b = 1;
p[n_, x_] := (a*x + b)^n
q[0, x_] := 1
q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]   (* A193820 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A128175 *)

From Peter Bala, Jul 16 2013: (Start)
T(n,k) = sum {i = 0..k} binomial(n-1,k-i) for 0 <= k <= n.
O.g.f.: (1 - x*t)^2/( (1 - 2*x*t)*(1 - (1 + x)*t) ) = 1 + (1 + x)*t + (1 + 2*x + 2*x^2)*t^2 + ....
The n-th row polynomial R(n,x) for n >= 1 is given by R(n,x) = 1/(1 - x)*( (x + 1)^(n-1) - 2^(n-1)*x^(n+1) ). Cf. A193823. (End)

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A055249 Triangle of partial row sums (prs) of triangle A055248 (prs of Pascal's triangle A007318).

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A052553 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.

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A102661 Triangle of partial sums of Stirling numbers of 2nd kind (A008277): T(n,k) = Sum_{i=1..k} Stirling2(n,i), 1<=k<=n.

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A090447 Triangle of partial products of binomials.

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A261363 Triangle read by rows: partial row sums of Sierpinski's triangle.

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A054124 Left Fibonacci row-sum array, n >= 0, 0<=k<=n.

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A163866 Partial sums of A007318.

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