A026729 -id:A026729 - cp's OEIS frontend

A213889 Triangle of coefficients of representations of columns of A213745 in binomial basis.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 0, 1, 5, 10, 10, 5, 1, 0, 0, 6, 15, 20, 15, 6, 1, 0, 0, 5, 21, 35, 35, 21, 7, 1, 0, 0, 4, 25, 56, 70, 56, 28, 8, 1, 0, 0, 3, 27, 80, 126, 126, 84, 36, 9, 1

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jun 23 2012

This array is the fifth array in the sequence of arrays A026729, A071675, A213887, A213888,..., such that the first two arrays are considered as triangles.

Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th row of the triangle. Then s_k(n)=sum{i=0,...,k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A213745. For example, s_1(n)=binomial(n,1)=n is the first column of A213745 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213745 for n>1, etc. In particular (see comment in A213745), in cases k=8,9 s_k(n) is A063417(n+2), A063418(n+2) respectively.

			As a triangle, this begins
n/k.|..0....1....2....3....4....5....6....7....8....9
=====================================================
.0..|..1
.1..|..0....1
.2..|..0....1....1
.3..|..0....1....2....1
.4..|..0....1....3....3....1
.5..|..0....1....4....6....4....1
.6..|..0....1....5...10...10....5....1
.7..|..0....0....6...15...20...15....6....1
.8..|..0....0....5...21...35...35...21....7....1
.9..|..0....0....4...25...56...70...56...28....8....1

Cf. A026729, A071675, A078803 (parts <=3), A213887 (parts <=4), A213888 (parts <=5).

Essentially the same as A061676.

pts := 6; # A213889 and A061676
g := 1/(1-t*z*add(z^i,i=0..pts-1)) ;
for n from 0 to 13 do
    for k from 0 to n do
        coeftayl(g,z=0,n) ;
        coeftayl(%,t=0,k) ;
        printf("%d ",%) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, May 28 2025

A064054 Tenth column of trinomial coefficients.

Original entry on oeis.org

5, 50, 266, 1016, 3139, 8350, 19855, 43252, 87802, 168168, 306735, 536640, 905658, 1481108, 2355962, 3656360, 5550755, 8260934, 12075184, 17363896, 24597925, 34370050, 47419905, 64662780, 87222720, 116470380, 154066125, 202008896, 262691396, 338962184

Offset: 0

Wolfdieter Lang, Aug 29 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

A005716 (ninth column), A111808.

A064054 := n -> GegenbauerC(`if`(9A064054(n)), n=5..20); # Peter Luschny, May 10 2016

Table[GegenbauerC[9, -n, -1/2], {n,5,50}] (* G. C. Greubel, Feb 28 2017 *)

for(n=0,25, print1(binomial(n+5,5)*(n^4 + 66*n^3 + 1307*n^2 + 8706*n + 15120) /(9!/5!), ", ")) \\ G. C. Greubel, Feb 28 2017

a(n) = A027907(n+5, 9).
a(n) = binomial(n+5, 5)*(n^4+66*n^3+1307*n^2+8706*n+15120) /(9!/5!).
G.f.: (1+x-x^2)*(5-5*x+x^2)/(1-x)^10, numerator polynomial is N3(9, x)= 5+0*x-9*x^2+6*x^3-x^4 from array A063420.
a(n) = A111808(n+5,9) for n>3. - Reinhard Zumkeller, Aug 17 2005
a(n) = 5*binomial(n+5,5) + 20*binomial(n+5,6) + 21*binomial(n+5,7) + 8*binomial(n+5,8) + binomial(n+5,9) (see our comment in A026729). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = GegenbauerC(N, -n, -1/2) where N = 9 if 9Peter Luschny, May 10 2016

A113953 A Jacobsthal triangle.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 4, 1, 0, 0, 4, 6, 1, 0, 0, 0, 12, 8, 1, 0, 0, 0, 8, 24, 10, 1, 0, 0, 0, 0, 32, 40, 12, 1, 0, 0, 0, 0, 16, 80, 60, 14, 1, 0, 0, 0, 0, 0, 80, 160, 84, 16, 1, 0, 0, 0, 0, 0, 32, 240, 280, 112, 18, 1, 0, 0, 0, 0, 0, 0, 192, 560, 448, 144, 20, 1, 0, 0, 0, 0, 0, 0, 64, 672, 1120, 672, 180, 22, 1

Offset: 0

Paul Barry, Nov 09 2005

Rows sums are the Jacobsthal numbers A001045(n+1).
Antidiagonal sums are the Padovan-Jacobsthal numbers A052947.
Inverse is (1,xc(-2x)), c(x) the g.f. of A000108, with general term k*C(2n-k-1,n-k)(-2)^(n - k)/n.
Triangle read by rows given by (0, 2, -2, 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 01 2013

			Rows begin
  1;
  0,  1;
  0,  2,  1;
  0,  0,  4,  1;
  0,  0,  4,  6,  1;
  0,  0,  0, 12,  8,  1;
  0,  0,  0,  8, 24, 10,  1;

D. Merlini, R. Sprugnoli, M. C. Verri, Strip tiling and regular grammars, Theor. Comp. Sci 242 (1-2) (2000) 109-124, Table 1, p=2.

A signed version is A110509.

G.f.: 1/(1-xy(1+2x)).
Riordan array (1, x(1+2x)).
T(n,k) = 2^(n-k)*binomial(k, n-k).
T(n,k) = A026729(n,k)*2^(n-k). - Philippe Deléham, Nov 22 2006
T(n,k) = T(n-1,k-1) + 2*T(n-2,k-1), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 01 2013

A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, -1, 3, -2, -1, 1, 0, -2, 5, -3, -1, 1, 1, -2, -2, 7, -4, -1, 1, -1, 5, -7, -1, 9, -5, -1, 1, 0, -3, 12, -15, 1, 11, -6, -1, 1, 1, -3, -3, 21, -26, 4, 13, -7, -1, 1, -1, 7, -15, 3, 31, -40, 8, 15, -8, -1, 1, 0, -4, 22, -42

Offset: 0

Philippe Deléham, Nov 12 2011

			Triangle begins :
1
-1, 1
0, -1, 1
1, -1, -1, 1
-1, 3, -2, -1, 1
0, -2, 5, -3, -1, 1
1, -2, -2, 7, -4, -1, 1
-1, 5, -7, -1, 9, -5, -1, 1

Cf. A026729, A063967, A129267, A176971 (diagonals sums).

T(n,k)=T(n-1,k-1)+T(n-2,k-1)-T(n-1,k)-T(n-2,k), T(0,0)=1.
G.f.: 1/(1-(y-1)*x-(y-1)*x^2).
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000748(n), A108520(n), A049347(n), A000007(n), A000045(n+1), A002605(n+1), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for x = -2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.

A099040 Riordan array (1, 2+2x).

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 0, 0, 8, 8, 0, 0, 4, 24, 16, 0, 0, 0, 24, 64, 32, 0, 0, 0, 8, 96, 160, 64, 0, 0, 0, 0, 64, 320, 384, 128, 0, 0, 0, 0, 16, 320, 960, 896, 256, 0, 0, 0, 0, 0, 160, 1280, 2688, 2048, 512, 0, 0, 0, 0, 0, 32, 960, 4480, 7168, 4608, 1024, 0, 0, 0, 0, 0, 0, 384, 4480, 14336, 18432, 10240, 2048

Offset: 0

Paul Barry, Sep 23 2004

Row sums give A002605. Diagonal sums give A052907.
The Riordan array (1,s+t*x) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).
T(n,k) is the number of compositions of n into two types of parts of size 1 and 2 that have exactly k parts. - Geoffrey Critzer, Aug 18 2012.
Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 22 2020

			Rows begin {1}, {0,2}, {0,2,4}, {0,0,8,8}, {0,0,4,24,16}, {0,0,0,24,64,32},...
T(3,2)=8 because we have: 1+2,1+2',1'+2,1'+2',2+1,2+1',2'+1,2'+1' where a part of the second type is designated by '. - _Geoffrey Critzer_, Aug 18 2012

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A002605, A026729, A038208, A052907.

nn = 8; CoefficientList[Series[1/(1 - 2 y x - 2 y x^2), {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 18 2012 *)

Number triangle T(n, k) = 2^k*binomial(k, n-k).
Columns have g.f. (2x+2x^2)^k.
T(n,k) = A026729(n,k)*2^k. - Philippe Deléham, Jul 28 2006
O.g.f.: 1/(1-2*y*x-2*y*x^2). - Geoffrey Critzer, Aug 18 2012.

A099093 Riordan array (1, 3+3x).

Original entry on oeis.org

1, 0, 3, 0, 3, 9, 0, 0, 18, 27, 0, 0, 9, 81, 81, 0, 0, 0, 81, 324, 243, 0, 0, 0, 27, 486, 1215, 729, 0, 0, 0, 0, 324, 2430, 4374, 2187, 0, 0, 0, 0, 81, 2430, 10935, 15309, 6561, 0, 0, 0, 0, 0, 1215, 14580, 45927, 52488, 19683, 0, 0, 0, 0, 0, 243, 10935, 76545, 183708, 177147, 59049

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A030195. Diagonal sums are A099094.
The Riordan array (1,s+tx) defines T(n,k) = binomial(k,n-k)s^k(t/s)^(n-k). The row sums satisfy a(n)=s*a(n-1)+t*a(n-2) and the diagonal sums satisfy a(n)=s*a(n-2)+t*a(n-3).
Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008

			Rows begin:
  1;
  0, 3;
  0, 3, 9;
  0, 0, 18, 27;
  0, 0, 9, 81, 81;
  0, 0, 0, 81, 324, 243;
  0, 0, 0, 27, 486, 1215, 729;
  ...

Cf. A038221.

[[Binomial(k,n-k)*3^k: k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Feb 21 2015 /* as the triangle */

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(binomial(k, n-k)*3^k, ", ");); print(););} \\ Michel Marcus, Feb 21 2015

T(n,k) = binomial(k, n-k)*3^k. - corrected by Michel Marcus, Feb 21 2015
Columns have g.f. (3x+3x^3)^k.
T(n,k) = A026729(n,k)*3^k. - Philippe Deléham, Jul 29 2006

A104730 Triangle read by rows: T(n,k)=C(n+1,k)-C(k,n-k+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 5, 10, 7, 1, 1, 6, 15, 19, 9, 1, 1, 7, 21, 35, 31, 11, 1, 1, 8, 28, 56, 69, 46, 13, 1, 1, 9, 36, 84, 126, 121, 64, 15, 1, 1, 10, 45, 120, 210, 251, 195, 85, 17, 1, 1, 11, 55, 165, 330, 462, 456, 295, 109, 19, 1, 1, 12, 66

Offset: 1

Gary W. Adamson, Mar 20 2005

Row sums are A027934: 1, 2, 5, 11, 24, 51, 107... Diagonal sums are A131298.

			The first few rows of the triangle are:
1;
1, 1;
1, 3, 1;
1, 4, 5, 1;
1, 5, 10, 7, 1;
1, 6, 15, 19, 9, 1;
1, 7, 31, 35, 31, 11, 1;
...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A027934, A026729, A007318.

Table[Binomial[n+1,k]-Binomial[k,n-k+1],{n,0,20},{k,0,n}]//Flatten (* Harvey P. Dale, Jan 16 2024 *)

Perform the operation A - B; then extract the triangle after deleting all zeros. P = infinite lower triangular Pascal's triangle matrix (A007318); B = A026729, as an infinite lower triangular matrix: [1; 0, 1;, 0, 1, 1; 0, 0, 2, 1; 0, 0, 1, 3, 1;...].

Better definition from Paul Barry, Jun 26 2007

More terms from Harvey P. Dale, Jan 16 2024

A104732 Square array T[i,j]=T[i-1,j]+T[i-1,j-1], T[1,j]=j, T[i,1]=1, read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 7, 8, 5, 1, 6, 9, 12, 12, 6, 1, 7, 11, 16, 20, 17, 7, 1, 8, 13, 20, 28, 32, 23, 8, 1, 9, 15, 24, 36, 48, 49, 30, 9, 1, 10, 17, 28, 44, 64, 80, 72, 38, 10, 1, 11, 19, 32, 52, 80, 112, 129, 102, 47, 11, 1, 12, 21, 36, 60, 96, 144, 192, 201, 140, 57, 12, 1

Offset: 1

Gary W. Adamson, Mar 20 2005

Original definition was "Triangle, row sums are A001924". Reading the rows of the triangle as antidiagonals of a square array allows a precise, yet simple definition and a method for computing the terms. - M. F. Hasler, Apr 26 2008

When formatted as a triangle, row sums are A001924: 1, 3, 7, 14, 26...(apply the partial sum operator twice to the Fibonacci sequence).

			The first few rows of the triangle (= rising diagonals of the square array) are:
1;
2, 1;
3, 3, 1;
4, 5, 4, 1;
5, 7, 8, 5, 1;
6, 9, 12, 12, 6, 1;
...

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001924, A026729.

A104732 := proc(i,j) coeftayl(coeftayl(x*y/(1-x)^2/(1-y*(1+x)),y=0,i),x=0,j) ; end: for d from 1 to 20 do for j from d to 1 by -1 do printf("%d,",A104732(d-j+1,j)) ; od: od: # R. J. Mathar, May 04 2008

nn = 10; Map[Select[#, # > 0 &] &,Drop[CoefficientList[
    Series[y x/(1 - x - y x + y x^3)/(1 - x), {x, 0, nn}], {x, y}],
1]] // Grid (* Geoffrey Critzer, Mar 17 2015 *)

def A104732_rows(n):
    """Produces n rows of A104732 triangle"""
    from operator import iadd
    a,b,c = [], [1], [1]
    for i in range(2,n+1):
            a,b = b, [i]+list(map(iadd,a,b[:-1]))+[1]
            c+=b
    return c
# Alec Mihailovs (alec(AT)mihailovs.com), May 04 2008

The triangle is extracted from A * B; where A = [1; 2, 1; 3, 2, 1;...], B = [1; 0, 1; 0, 1, 1; 0, 0, 2, 1;...]; both infinite lower triangular matrices with the rest of the terms zeros. The sequence in "B" (1, 0, 1, 0, 1, 1, 0, 0, 2, 1...) = A026729.

As a square array, g.f. Sum T[i,j] x^j y^i = xy/((1-(1+x)y)*(1-x)^2). - Alec Mihailovs (alec(AT)mihailovs.com), Apr 26 2008

Edited by M. F. Hasler, Apr 26 2008

More terms from R. J. Mathar and Alec Mihailovs (alec(AT)mihailovs.com), May 04 2008

A104734 Triangle T(n,k) = sum_{j=k..n} (2n-2j+1)*binomial(k,j-k), read by rows, 0<=k<=n.

Original entry on oeis.org

1, 3, 1, 5, 4, 1, 7, 8, 5, 1, 9, 12, 12, 6, 1, 11, 16, 20, 17, 7, 1, 13, 20, 28, 32, 23, 8, 1, 15, 24, 36, 48, 49, 30, 9, 1, 17, 28, 44, 64, 80, 72, 38, 10, 1, 19, 32, 52, 80, 112, 129, 102, 47, 11, 1, 21, 36, 60, 96, 144, 192, 201, 140, 57, 12, 1, 23, 40, 68, 112, 176, 256, 321, 303, 187, 68, 13, 1, 25, 44, 76, 128, 208, 320, 448, 522, 443, 244, 80, 14, 1

Offset: 0

Gary W. Adamson, Mar 20 2005

Array A210489 (without first row) read downwards antidiagonals. - R. J. Mathar, Sep 17 2013

			First few rows of the triangle are:
1;
3, 1;
5, 4, 1;
7, 8, 5, 1;
9, 12, 12, 6, 1;
11, 16, 20, 17, 7, 1;
...

Cf. A001891 (row sums), A026729.

Matrix product of the triangle A = A099375 by B = [1; 0, 1; 0, 1, 1; 0, 0, 2, 1; 0, 0, 1, 3, 1;...] (which is the triangular view of A026729).

A124369 Riordan array (1/((1-x-x^2)(1+x+x^2)),x(1+x)/((1-x-x^2)(1+x+x^2))).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 2, 6, 4, 3, 1, 4, 9, 12, 7, 4, 1, 7, 17, 24, 21, 11, 5, 1, 10, 34, 48, 50, 34, 16, 6, 1, 17, 58, 103, 110, 91, 52, 22, 7, 1, 28, 104, 200, 250, 220, 152, 76, 29, 8, 1, 44, 188, 385, 534, 530, 400, 239, 107, 37, 9, 1

Offset: 0

Paul Barry, Oct 27 2006

Row sums are A123392. Diagonal sums are A124370. First column is A094686. Product of A026729 and abs(A049310).

			Triangle begins
1,
0, 1,
1, 1, 1,
2, 2, 2, 1,
2, 6, 4, 3, 1,
4, 9, 12, 7, 4, 1,
7, 17, 24, 21, 11, 5, 1,
10, 34, 48, 50, 34, 16, 6, 1,
17, 58, 103, 110, 91, 52, 22, 7, 1

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

T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + 2*T(n-3,k) + T(n-4,k), T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 22 2014

cp's OEIS Frontend

A213889 Triangle of coefficients of representations of columns of A213745 in binomial basis.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

A064054 Tenth column of trinomial coefficients.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A113953 A Jacobsthal triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Examples

Crossrefs

Formula

A099040 Riordan array (1, 2+2x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A099093 Riordan array (1, 3+3x).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

PARI

Formula

A104730 Triangle read by rows: T(n,k)=C(n+1,k)-C(k,n-k+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A104732 Square array T[i,j]=T[i-1,j]+T[i-1,j-1], T[1,j]=j, T[i,1]=1, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs