A097806 -id:A097806 - cp's OEIS frontend

A167366 Triangle read by rows, 2*A047999 - A097806 (signed) = twice Sierpinski's gasket - the signed pair sum operator.

1, 3, 1, 2, 1, 1, 2, 2, 3, 1, 2, 0, 0, 1, 1, 2, 2, 0, 0, 3, 1, 2, 0, 2, 0, 2, 1, 1, 2, 2, 2, 2, 2, 2, 3, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 3, 1, 2, 0, 2, 0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 2, 2, 0, 0, 0, 0, 2, 2, 3, 1, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 1, 1

Offset: 0

Gary W. Adamson & Mats Granvik, Nov 01 2009

Row sums = A167275: (1, 4, 4, 8, 4, 8, 8, 16,...).

			First few rows of the triangle =
1;
3, 1;
2, 1, 1;
2, 2, 3, 1;
2, 0, 0, 1, 1;
2, 2, 0, 0, 3, 1;
2, 0, 2, 0, 2, 1, 1;
2, 2, 2, 2, 2, 2, 3, 1;
2, 0, 0, 0, 0, 0, 0, 1, 1;
2, 2, 0, 0, 0, 0, 0, 0, 3, 1;
2, 0, 2, 0, 0, 0, 0, 0, 2, 1, 1;
2, 2, 2, 2, 0, 0, 0, 0, 2, 2, 3, 1;
2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 1, 1;
2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 3, 1;
2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 1, 1;
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1;
2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1;
2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1;
2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 1;
...

Cf. A047999, A097806

Triangle read by rows, 2*A047999 - A096806, the pair sum operator.
A096806 is signed, rightmost diagonal = (+,+,+,...); adjacent diagonal is
signed (-,-,-,...).

A122908 A central binomial scaling of the Riordan array (1+x,x) (A097806).

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 0, 3, 6, 0, 0, 0, 0, 6, 10, 0, 0, 0, 0, 0, 10, 20, 0, 0, 0, 0, 0, 0, 20, 35, 0, 0, 0, 0, 0, 0, 0, 35, 70, 0, 0, 0, 0, 0, 0, 0, 0, 70, 126, 0, 0, 0, 0, 0, 0, 0, 0, 0, 126, 252

Offset: 0

Paul Barry, Sep 18 2006

Row sums are A050168. Diagonal sums are A001045 doubled. Row sums of inverse are 1/C(2n,n) aerated. Applications to other sequences are obvious.

			Triangle begins
.1,
.1, 1,
.0, 1, 2,
.0, 0, 2, 3,
.0, 0, 0, 3, 6,
.0, 0, 0, 0, 6, 10,
.0, 0, 0, 0, 0, 10, 20,
.0, 0, 0, 0, 0, 0, 20, 35,
.0, 0, 0, 0, 0, 0, 0, 35, 70,
.0, 0, 0, 0, 0, 0, 0, 0, 70, 126,
.0, 0, 0, 0, 0, 0, 0, 0, 0, 126, 252

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

A128151 A002260 * A097806.

Original entry on oeis.org

1, 3, 2, 3, 5, 3, 3, 5, 7, 4, 3, 5, 7, 9, 5, 3, 5, 7, 9, 11, 6, 3, 5, 7, 9, 11, 13, 7, 3, 5, 7, 9, 11, 13, 15, 8, 3, 5, 7, 9, 11, 13, 15, 17, 9, 3, 5, 7, 9, 11, 13, 15, 17, 19, 10

Offset: 1

Gary W. Adamson, Feb 16 2007

Row sums = A028387: (1, 5, 11, 19, 29, 41, ...).

			First few rows of the triangle:
  1;
  3, 2;
  3, 5, 3;
  3, 5, 7, 4;
  3, 5, 7, 9, 5;
  ...

Cf. A002260, A097806, A028387.

A002260 * A097806 as infinite lower triangular matrices. a(1) = 1; n-th row has (n-1) terms in the sequence (3, 5, 7, 9, ...) followed by "n".

A128185 A097806 * A051731.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 1, 0, 1, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Feb 17 2007

A128184 = A051731 * A097806 Row sums = 1, 3, 4, 5, 5, 6, 6, 6, 7, 7, ... (the first 10 terms of A108852).

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  2, 1, 1, 1;
  2, 1, 0, 1, 1;
  2, 1, 1, 0, 1, 1;
  2, 1, 1, 0, 0, 1, 1;
  2, 1, 0, 1, 0, 0, 1, 1;
  ...

Cf. A128184, A097806, A051731, A108852.

A097806 * A051731 as infinite lower triangular matrices.

A130453 A097806 * A059268.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 2, 4, 8, 8, 2, 4, 8, 16, 16, 2, 4, 8, 16, 32, 32, 2, 4, 8, 16, 32, 64, 64, 2, 4, 8, 16, 32, 64, 128, 128

Offset: 1

Gary W. Adamson, May 26 2007

Row sums = A033484: (1, 4, 10, 22, 46, 94, ...).

A130459 = A059268 * A097806.

			First few rows of the triangle:
  1;
  2, 2;
  2, 4, 4;
  2, 4, 8,  8;
  2, 4, 8, 16, 16;
  2, 4, 8, 16, 32, 32;
  ...

Cf. A097806, A059268, A033484, A130459.

A097806 * A059268 as infinite lower triangular matrices.

A131398 3*A007318 - A097806 - A000012.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 2, 8, 7, 1, 2, 11, 17, 10, 1, 2, 14, 29, 29, 13, 1, 2, 17, 44, 59, 44, 16, 1, 2, 20, 62, 104, 104, 62, 19, 1, 2, 23, 83, 167, 209, 167, 83, 22, 1, 2, 26, 107, 251, 377, 377, 251, 107, 25, 1

Offset: 0

Gary W. Adamson, Jul 05 2007

Row sums = A077802, (1, 2, 7, 18, 41, 88,...).

			First few rows of the triangle are:
1;
1, 1;
2, 4, 1;
2, 8, 7, 1;
2, 11, 17, 10, 1;
2, 14, 29, 29, 13, 1;
...

Cf. A007318, A097806, A000012, A077802.

3*A007318 - A097806 - A000012 as infinite lower triangular matrices.

A133807 A007318 * (A097806 + A133566 - I), where I is the identity matrix.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 4, 1, 5, 16, 10, 6, 1, 6, 25, 20, 20, 6, 1, 7, 36, 35, 50, 21, 8, 1, 8, 49, 56, 105, 56, 35, 8, 1, 9, 64, 84, 196, 126, 112, 36, 10, 1, 10, 81, 120, 336, 252, 294, 120, 54, 10, 1

Offset: 1

Gary W. Adamson, Sep 23 2007

Row sums = A133806: (1, 3, 8, 18, 38, 78, 158, ...).

			First few rows of the triangle:
  1;
  2,  1;
  3,  4,  1;
  4,  9,  4,  1;
  5, 16, 10,  6,  1;
  6, 25, 20, 20,  6,  1;
  7, 36, 35, 50, 21,  8,  1;
  ...

Cf. A133566, A007318 , A097806, A133806.

Binomial transform of matrix M, where M = (A097806 + A133566 - I) = triangle with (1,1,1,...) in the main diagonal, (1,2,1,2,1,...) in the subdiagonal and the rest zeros. I = Identity matrix.

A046854 Triangle read by rows: T(n, k) = binomial(floor((n+k)/2), k), n >= k >= 0.

Original entry on oeis.org

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

Offset: 0

Wouter Meeussen

Row sums are Fibonacci(n+2). Diagonal sums are A016116. - Paul Barry, Jul 07 2004
Riordan array (1/(1-x), x/(1-x^2)). Matrix inverse is A106180. - Paul Barry, Apr 24 2005
As an infinite lower triangular matrix * [1,2,3,...] = A055244. - Gary W. Adamson, Dec 23 2008
From Emeric Deutsch, Jun 18 2010: (Start)
T(n,k) is the number of alternating parity increasing subsequences of {1,2,...,n} of size k, starting with an odd number (Terquem's problem, see the Riordan reference, p. 17). Example: T(8,5)=6 because we have 12345, 12347, 12367, 12567, 14567, and 34567.
T(n,k) is the number of alternating parity increasing subsequences of {1,2,...,n,n+1} of size k, starting with an even number. Example: T(7,4)=5 because we have 2345, 2347, 2367, 2567, and 4567. (End)
From L. Edson Jeffery, Mar 01 2011: (Start)
This triangle can be constructed as follows. Interlace two copies of the table of binomial coefficients to get the preliminary table
  1
  1
  1  1
  1  1
  1  2  1
  1  2  1
  1  3  3  1
  1  3  3  1
  ...,
then shift each entire r-th column up r rows, r=0,1,2,.... Also, a signed version of this sequence (A187660 in tabular form) begins with
  1;
  1, -1;
  1, -1, -1;
  1, -2, -1, 1;
  1, -2, -3, 1, 1;
  ...
(compare with A066170, A130777). Let T(N,k) denote the k-th entry in row N of the signed table. Then, for N>1, row N gives the coefficients of the characteristic function p_N(x) = Sum_{k=0..N} T(N,k)*x^(N-k) = 0 of the N X N matrix U_N=[(0 ... 0 1);(0 ... 0 1 1);...;(0 1 ... 1);(1 ... 1)]. Now let Q_r(t) be a polynomial with recurrence relation Q_r(t)=t*Q_(r-1)(t)-Q_(r-2)(t) (r>1), with Q_0(t)=1 and Q_1(t)=t. Then p_N(x)=0 has solutions Q_(N-1)(phi_j), where phi_j=2*(-1)^(j-1)*cos(j*Pi/(2*N+1)), j=1,2,...,N.
For example, row N=3 is {1,-2,-1,1}, giving the coefficients of the characteristic function p_3(x) = x^3-2*x^2-x+1 = 0 for the 3 X 3 matrix U_3=[(0 0 1);(0 1 1);(1 1 1)], with eigenvalues Q_2(phi_j)=[2*(-1)^(j-1)*cos(j*Pi/7)]^2-1, j=1,2,3. (End)
Given the signed polynomials (+--++--,...) of the triangle, the largest root of the n-th row polynomial is the longest (2n+1) regular polygon diagonal length, with edge = 1. Example: the largest root to x^3 - 2x^2 - x + 1 = 0 is 2.24697...; the longest heptagon diagonal, sin(3*Pi/7)/sin(Pi/7). - Gary W. Adamson, Sep 06 2011
Given the signed polynomials from Gary W. Adamson's comment, the largest root of the n-th polynomial also equals the length from the center to a corner (vertex) of a regular 2*(2*n+1)-sided polygon with side (edge) length = 1. - L. Edson Jeffery, Jan 01 2012
Put f(x,0) = 1 and f(x,n) = x + 1/f(x,n-1). Then f(x,n) = u(x,n)/v(x,n), where u(x,n) and v(x,n) are polynomials. The flattened triangles of coefficients of u and v are both essentially A046854, as indicated by the Mathematica program headed "Polynomials". - Clark Kimberling, Oct 12 2014
From Jeremy Dover, Jun 07 2016: (Start)
T(n,k) is the number of binary strings of length n+1 starting with 0 that have exactly k pairs of consecutive 0's and no pairs of consecutive 1's.
T(n,k) is the number of binary strings of length n+2 starting with 1 that have exactly k pairs of consecutive 0's and no pairs of consecutive 1's. (End)

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

J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, 1978. [Emeric Deutsch, Jun 18 2010]

Nathaniel Johnston, Rows 0..100, flattened
Robert G. Donnelly, Molly W. Dunkum, and Rachel McCoy, Olry Terquem's forgotten problem, arXiv:2303.05949 [math.HO], 2023.
Jeremy M. Dover, Some Notes on Pairs in Binary Strings, arXiv:1609.00980 [math.CO], 2016.
Dominique Foata and Guo-Niu Han, Multivariable Tangent and Secant q-derivative Polynomials, arXiv:1304.2486 [math.CO], 2013; see Fig. 10.1.
Index entries for triangles and arrays related to Pascal's triangle

Reflected version of A065941, which is considered the main entry. A deficient version is in A030111.
Cf. A066170, A097806, A134513, A168561, A187660.
Cf. A055244. - Gary W. Adamson, Dec 23 2008

Flat(List([0..16], n-> List([0..n], k-> Binomial(Int((n+k)/2), k) ))); # G. C. Greubel, Jul 13 2019

a046854 n k = a046854_tabl !! n !! k
a046854_row n = a046854_tabl !! n
a046854_tabl = [1] : f [1] [1,1] where
   f us vs = vs : f vs  (zipWith (+) (us ++ [0,0]) ([0] ++ vs))
-- Reinhard Zumkeller, Apr 24 2013

[Binomial(Floor((n+k)/2), k): k in [0..n], n in [0..16]]; // G. C. Greubel, Jul 13 2019

A046854:= proc(n,k): binomial(floor(n/2+k/2), k) end: seq(seq(A046854(n,k),k=0..n),n=0..16); # Nathaniel Johnston, Jun 30 2011

Table[Binomial[Floor[(n+k)/2], k], {n,0,16}, {k,0,n}]//Flatten
(* next program: Polynomials *)
z = 12; f[x_, n_] := x + 1/f[x, n - 1]; f[x_, 1] = 1;
t = Table[Factor[f[x, n]], {n, 1, z}]
u = Flatten[CoefficientList[Numerator[t], x]] (* this sequence *)
v = Flatten[CoefficientList[Denominator[t], x]]
(* Clark Kimberling, Oct 13 2014 *)

T(n,k) = binomial((n+k)\2, k); \\ G. C. Greubel, Jul 13 2019

[[binomial(floor((n+k)/2), k) for k in (0..n)] for n in (0..16)] # G. C. Greubel, Jul 13 2019

T(n,k) = binomial(floor((n+k)/2), k).
G.f.: (1+x)/(1-x*y-x^2). - Ralf Stephan, Feb 13 2005
Triangle = A097806 * A168561, as infinite lower triangular matrices. - Gary W. Adamson, Oct 28 2007
T(n,k) = A065941(n,n-k) = abs(A130777(n,k)) = abs(A066170(n,k)) = abs(A187660(n,k)). - Johannes W. Meijer, Aug 08 2011
For n > 1: T(n, k) = T(n-1, k-1) + T(n-2, k), 0 < k < n-1. - Reinhard Zumkeller, Apr 24 2013
T(n,k) = A168561(n,k) + A168561(n-1,k). - R. J. Mathar, Feb 10 2024

A029653 Numbers in (2,1)-Pascal triangle (by row).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 7, 9, 5, 1, 2, 9, 16, 14, 6, 1, 2, 11, 25, 30, 20, 7, 1, 2, 13, 36, 55, 50, 27, 8, 1, 2, 15, 49, 91, 105, 77, 35, 9, 1, 2, 17, 64, 140, 196, 182, 112, 44, 10, 1, 2, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 2, 21, 100, 285

Offset: 0

Mohammad K. Azarian

Reverse of A029635. Row sums are A003945. Diagonal sums are Fibonacci(n+2) = Sum_{k=0..floor(n/2)} (2n-3k)*C(n-k,n-2k)/(n-k). - Paul Barry, Jan 30 2005
Riordan array ((1+x)/(1-x), x/(1-x)). The signed triangle (-1)^(n-k)T(n,k) or ((1-x)/(1+x), x/(1+x)) is the inverse of A055248. Row sums are A003945. Diagonal sums are F(n+2). - Paul Barry, Feb 03 2005
Row sums = A003945: (1, 3, 6, 12, 24, 48, 96, ...) = (1, 3, 7, 15, 31, 63, 127, ...) - (0, 0, 1, 3, 7, 15, 31, ...); where (1, 3, 7, 15, ...) = A000225. - Gary W. Adamson, Apr 22 2007
Triangle T(n,k), read by rows, given by (2,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 17 2011
A029653 is jointly generated with A208510 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+x*v(n-1)x and v(n,x)=u(n-1,x)+x*v(n-1,x)+1. See the Mathematica section. - Clark Kimberling, Feb 28 2012
For a closed-form formula for arbitrary left and right borders of Pascal like triangle, see A228196. - Boris Putievskiy, Aug 18 2013
For a closed-form formula for generalized Pascal's triangle, see A228576. - Boris Putievskiy, Sep 04 2013
The n-th row polynomial is (2 + x)*(1 + x)^(n-1) for n >= 1. More generally, the n-th row polynomial of the Riordan array ( (1-a*x)/(1-b*x), x/(1-b*x) ) is (b - a + x)*(b + x)^(n-1) for n >= 1. - Peter Bala, Feb 25 2018

			The triangle T(n,k) begins:
n\k 0  1  2   3   4   5   6   7  8  9 10 ...
0:  1
1:  2  1
2:  2  3  1
3:  2  5  4   1
4:  2  7  9   5   1
5:  2  9 16  14   6   1
6:  2 11 25  30  20   7   1
7:  2 13 36  55  50  27   8   1
8:  2 15 49  91 105  77  35   9  1
9:  2 17 64 140 196 182 112  44 10  1
10: 2 19 81 204 336 378 294 156 54 11  1
... Reformatted. - _Wolfdieter Lang_, Jan 09 2015
With the array M(k) as defined in the Formula section, the infinite product M(0)*M(1)*M(2)*... begins
/1        \/1         \/1        \      /1        \
|2 1      ||0 1       ||0 1      |      |2 1      |
|2 1 1    ||0 2 1     ||0 0 1    |... = |2 3 1    |
|2 1 1 1  ||0 2 1 1   ||0 0 2 1  |      |2 5 4 1  |
|2 1 1 1 1||0 2 1 1 1 ||0 0 2 1 1|      |2 7 9 5 1|
|...      ||...       ||...      |      |...      |
- _Peter Bala_, Dec 27 2014

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
Hacene Belbachir and Athmane Benmezai, Expansion of Fibonacci and Lucas Polynomials: An Answer to Prodinger's Question, Journal of Integer Sequences, Vol. 15 (2012), #12.7.6.
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 39.
H. Hosoya, Pascal's triangle, non-adjacent numbers and D-dimensional atomic orbitals, J. Math. Chemistry, vol. 23, 1998, 169-178.
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
M. Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 8.
Mark C. Wilson, Asymptotics for generalized Riordan arrays. International Conference on Analysis of Algorithms DMTCS proc. AD. Vol. 323. 2005.

(d, 1) Pascal triangles: A007318(d=1), A093560(3), A093561(4), A093562(5), A093563(6), A093564(7), A093565(8), A093644(9), A093645(10).
Cf. A003945, A208510, A228196, A228576.
Cf. A078812, A106195.

a029653 n k = a029653_tabl !! n !! k
a029653_row n = a029653_tabl !! n
a029653_tabl = [1] : iterate
               (\xs -> zipWith (+) ([0] ++ xs) (xs ++ [0])) [2, 1]
-- Reinhard Zumkeller, Dec 16 2013

A029653 :=  proc(n,k)
if n = 0 then
  1;
else
  binomial(n-1, k)+binomial(n, k)
fi
end proc: # R. J. Mathar, Jun 30 2013

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A208510 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A029653 *)
(* Clark Kimberling, Feb 28 2012 *)

from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1
def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017

from math import comb, isqrt
def A029653(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),a:=n-comb(r+1,2))*((r<<1)-a)//r if n else 1 # Chai Wah Wu, Nov 12 2024

T(n, k) = C(n-2, k-1) + C(n-2, k) + C(n-1, k-1) + C(n-1, k) except for n=0.
G.f.: (1 + x + y + xy)/(1 - y - xy). - Ralf Stephan, May 17 2004
T(n, k) = (2n-k)*binomial(n, n-k)/n, n, k > 0. - Paul Barry, Jan 30 2005
Sum_{k=0..n} T(n, k)*x^k gives A003945-A003954 for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. - Philippe Deléham, Jul 10 2005
T(n, k) = C(n-1, k) + C(n, k). - Philippe Deléham, Jul 10 2005
Equals A097806 * A007318, i.e., the pairwise operator * Pascal's Triangle as infinite lower triangular matrices. - Gary W. Adamson, Apr 22 2007
From Peter Bala, Dec 27 2014: (Start)
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(2 + 5*x + 4*x^2/2! + x^3/3!) = 2 + 7*x + 16*x^2/2! + 30*x^3/3! + 50*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ).
Let M denote the lower unit triangular array with 1's on the main diagonal and 1's everywhere else below the main diagonal except for the first column which consists of the sequence [1,2,2,2,...]. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)

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cp's OEIS Frontend

A134312 A097806 * A134309.

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A167366 Triangle read by rows, 2*A047999 - A097806 (signed) = twice Sierpinski's gasket - the signed pair sum operator.

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A122908 A central binomial scaling of the Riordan array (1+x,x) (A097806).

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A128151 A002260 * A097806.

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A128185 A097806 * A051731.

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A130453 A097806 * A059268.

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A131398 3*A007318 - A097806 - A000012.

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A133807 A007318 * (A097806 + A133566 - I), where I is the identity matrix.

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A046854 Triangle read by rows: T(n, k) = binomial(floor((n+k)/2), k), n >= k >= 0.

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