A213888 -id:A213888 - cp's OEIS frontend

A213887 Triangle of coefficients of representations of columns of A213743 in binomial basis.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 0, 4, 6, 4, 1, 0, 0, 3, 10, 10, 5, 1, 0, 0, 2, 12, 20, 15, 6, 1, 0, 0, 1, 12, 31, 35, 21, 7, 1, 0, 0, 0, 10, 40, 65, 56, 28, 8, 1, 0, 0, 0, 6, 44, 101, 120, 84, 36, 9, 1, 0

Offset: 0

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

This triangle is the third array in the sequence of arrays A026729, A071675 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 A213743. For example, s_1(n)=binomial(n,1)=n is the first column of A213743 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213743 for n>1, etc. In particular (see comment in A213743), in cases k=6,7,8,9  s_k(n) is A064056(n+2), A064057(n+2), A064058(n+2), A000575(n+3) respectively.
Riordan array (1,x+x^2+x^3+x^4). A186332 with additional 0 column. - Ralf Stephan, Dec 31 2013

			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....0....4....6....4....1
.6..|..0....0....3...10...10....5....1
.7..|..0....0....2...12...20...15....6....1
.8..|..0....0....1...12...31...35...21....7....1
.9..|..0....0....0...10...40...65...56...28....8....1

Cf. A026729, A071675, A030528 (parts <=2), A078803 (parts <=3), A213888 (parts <=5), A061676 and A213889 (parts <=6).

pts := 4; # A213887
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

A078803 Triangular array T given by T(n,k) = number of compositions of n into k parts, each in the set {1,2,3}.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 0, 3, 3, 1, 0, 2, 6, 4, 1, 0, 1, 7, 10, 5, 1, 0, 0, 6, 16, 15, 6, 1, 0, 0, 3, 19, 30, 21, 7, 1, 0, 0, 1, 16, 45, 50, 28, 8, 1, 0, 0, 0, 10, 51, 90, 77, 36, 9, 1, 0, 0, 0, 4, 45, 126, 161, 112, 45, 10, 1, 0, 0, 0, 1, 30, 141, 266, 266, 156, 55, 11, 1, 0, 0, 0, 0, 15, 126

Offset: 1

Clark Kimberling, Dec 06 2002

Number of lattice paths from (0,0) to (n,k) using steps (1,1), (2,1), (3,1). - Joerg Arndt, Jul 05 2011
Reversing the rows produces A078802. Row sums: A000073.
Number of tribonacci binary words of length n-1 having k-1 1's. A tribonacci binary word is a binary word having no three consecutive 0's. Example: T(6,3)=7 because we have 00101,00110,01001,01010,01100,10010 and 10100. - Emeric Deutsch, Jun 16 2007
This is the Riordan array (1,x+x^2+x^3)(A071675) without its column k=0. - Vladimir Kruchinin, Feb 10 2011

			T(5,2) = 2 counts the compositions 2+3 and 3+2.
Triangle begins
  1;
  1, 1;
  1, 2, 1;
  0, 3, 3, 1;
  0, 2, 6, 4, 1;
  0, 1, 7, 10, 5, 1;
  0, 0, 6, 16, 15, 6, 1;
  0, 0, 3, 19, 30, 21, 7, 1;
  0, 0, 1, 16, 45, 50, 28, 8, 1;
  0, 0, 0, 10, 51, 90, 77, 36, 9, 1;
  0, 0, 0, 4, 45, 126, 161, 112, 45, 10, 1;
  0, 0, 0, 1, 30, 141, 266, 266, 156, 55, 11, 1;

Clark Kimberling, Binary words with restricted repetitions and associated compositions of integers, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 141-151.

Alois P. Heinz, Rows n = 1..141, flattened
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 18.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Cf. A027907, A078802, A030528 (parts <=2), A213887 (parts <=4), A213888 (parts <=5), A061676 and A213889 (parts <=6).

A078803 := proc(n,k) add( binomial(j,n-3*k+2*j)*binomial(k,j),j=0..k) ; end proc:
# R. J. Mathar, Feb 22 2011

nn=8;CoefficientList[Series[1/(1-y(x+x^2+x^3)),{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Jan 08 2013 *)

T(n, k) = t(n-1, n-k), for 1<=k<=n, for n>=1, where the array t is given by A078802.
G.f.: 1/(1-t*z*(1+z+z^2))-1. - Emeric Deutsch, Mar 10 2004
T(n,k) = Sum_{j=0..k} C(j,n-3*k+2*j)*C(k,j). - Vladimir Kruchinin, Feb 10 2011

More terms from Emeric Deutsch, Jun 16 2007

A061676 Triangle T(n,k) of number of ways of throwing k standard dice to produce a total of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 0, 6, 15, 20, 15, 6, 1, 0, 5, 21, 35, 35, 21, 7, 1, 0, 4, 25, 56, 70, 56, 28, 8, 1, 0, 3, 27, 80, 126, 126, 84, 36, 9, 1, 0, 2, 27, 104, 205, 252, 210, 120, 45, 10, 1, 0, 1, 25, 125, 305, 456, 462, 330, 165, 55, 11, 1

Offset: 1

Henry Bottomley, Apr 01 2002

			Rows start:
1;
1,1;
1,2,1;
1,3,3,1;
1,4,6,4,1;
1,5,10,10,5,1;
0,6,15,20,15,6,1;
0,5,21,35,35,21,7,1;
etc.
T(8,2)=5 since 8 =2+6 =3+5 =4+4 =5+3 =6+2.

First 21 terms as A007318 (see formula). Cf. A001592, A069713.

Cf. A030528 (2-sided dice), A078803 (3-sided), A213887 (4-sided), A213888 (5-sided).

pts := 6; # A213889 and A061676
g := 1/(1-t*z*add(z^i,i=0..pts-1)) ;
for n from 1 to 13 do
    for k from 1 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

T(n, k)=T(n-1, k-1)+T(n-2, k-1)+T(n-3, k-1)+T(n-4, k-1)+T(n-5, k-1)+T(n-6, k-1) starting with T(0, 0)=1. T(n, k)=T(7k-n, k); if n>6k or n6k-6, T(n, k)=C(7k-n-1, k-1); T([7k/2], k)=A018901(k).

A063262 Eighth column (k=7) of sextinomial array A063260.

Original entry on oeis.org

4, 27, 104, 305, 756, 1667, 3368, 6354, 11340, 19327, 31680, 50219, 77324, 116055, 170288, 244868, 345780, 480339, 657400, 887589, 1183556, 1560251, 2035224, 2628950, 3365180, 4271319, 5378832

Offset: 0

Wolfdieter Lang, Jul 24 2001

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A062989, A063260, A063261, A063263, A063264, A213888.

CoefficientList[Series[(4-5x+5x^3-4x^4+x^5)/(1-x)^8,{x,0,30}],x] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{4,27,104,305,756,1667,3368,6354},30] (* Harvey P. Dale, Mar 07 2023 *)

a(n) = A063260(n+2, 7 )= (n+1)*(n+2)*(n^5+32*n^4+413*n^3+2722*n^2+9432*n+10080)/7!.
G.f.: (4-5*x+5*x^3-4*x^4+x^5)/(1-x)^8; the numerator polynomial is N6(7, x) from row n=7 of array A063261.
a(n) = 4*C(n+2,2) + 15*C(n+2,3) + 20*C(n+2,4) + 15*C(n+2,5) + 6*C(n+2,6) + C(n+2,7) (see comment in A213888). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A063263 Ninth column (k=8) of sextinomial array A063260.

Original entry on oeis.org

3, 27, 125, 420, 1161, 2807, 6147, 12465, 23760, 43032, 74646, 124787, 202020, 317970, 488138, 732870, 1078497, 1558665, 2215875, 3103254, 4286579, 5846577, 7881525, 10510175, 13875030, 18145998, 23524452

Offset: 0

Wolfdieter Lang, Jul 24 2001

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A062989, A063260, A063261, A063262, A063264, A213888.

a(n) = A063260(n+2, 8) = (n+1)*(n+2)*(n+3)*(n^5+38*n^4+587*n^3+4678*n^2+19896*n+20160)/8!.
G.f.: (3-10*x^2+15*x^3-9*x^4+2*x^5)/(1-x)^9; the numerator polynomial is N6(8, x) from row n=8 of array A063261.
a(n) = 3*C(n+2,2) + 18*C(n+2,3) + 35*C(n+2,4) + 35*C(n+2,5) + 21*C(n+2,6) + 7*C(n+2,7) + C(n+2,8) (see comment in A213888). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A063264 Tenth column (k=9) of sextinomial array A063260.

Original entry on oeis.org

2, 25, 140, 540, 1666, 4417, 10480, 22825, 46420, 89232, 163592, 288015, 489580, 806990, 1294448, 2026502, 3104030, 4661555, 6876100, 9977814, 14262622, 20107175, 27986400, 38493975, 52366080, 70508802

Offset: 0

Wolfdieter Lang, Jul 24 2001

Ray Chandler, 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).

Cf. A062989, A063260, A063261, A063262, A063263, A213888.

a(n) = A063260(n+2, 9) = (n+1)*(n+2)*(n+3)*(n+4)*(n^5+44*n^4+791*n^3+7384*n^2+37140*n+30240)/9!.
G.f.: (2+5*x-20*x^2+25*x^3-14*x^4+3*x^5)/(1-x)^10; the numerator polynomial is N6(8, x) from row n=8 of array A063261.
a(n) = 2*C(n+2,2) + 19*C(n+2,3) + 52*C(n+2,4) + 70*C(n+2,5) + 56*C(n+2,6) + 28*C(n+2,7) + 8*C(n+2,8) + C(n+2,9) (see comment in A213888). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

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

Original entry on oeis.org

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

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A213887 Triangle of coefficients of representations of columns of A213743 in binomial basis.

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A078803 Triangular array T given by T(n,k) = number of compositions of n into k parts, each in the set {1,2,3}.

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A061676 Triangle T(n,k) of number of ways of throwing k standard dice to produce a total of n.

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A063262 Eighth column (k=7) of sextinomial array A063260.

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A063263 Ninth column (k=8) of sextinomial array A063260.

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A063264 Tenth column (k=9) of sextinomial array A063260.

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

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