A213887 -id:A213887 - cp's OEIS frontend

A213888 Triangle of coefficients of representations of columns of A213744 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, 0, 5, 10, 10, 5, 1, 0, 0, 4, 15, 20, 15, 6, 1, 0, 0, 3, 18, 35, 35, 21, 7, 1, 0, 0, 2, 19, 52, 70, 56, 28, 8, 1, 0, 0, 1, 18, 68, 121, 126, 84, 36, 9, 1, 0

Offset: 0

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

This triangle is the fourth array in the sequence of arrays A026729, A071675, A213887,..., 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 A213744. For example, s_1(n)=binomial(n,1)=n is the first column of A213744 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213744 for n>1, etc. In particular (see comment inA213744), in cases k=7,8,9 s_k(n) is A063262(n+2), A063263(n+2), A063264(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....0....5...10...10....5....1
.7..|..0....0....4...15...20...15....6....1
.8..|..0....0....3...18...35...35...21....7....1
.9..|..0....0....2...19...52...70...56...28....8....1

Cf. A026729, A071675, A213887, A030528 (parts <=2), A078803 (parts <=3), A213887 (parts <=4).

pts := 5; # A213888
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

A000575 Tenth column of quintinomial coefficients.

Original entry on oeis.org

10, 80, 365, 1246, 3535, 8800, 19855, 41470, 81367, 151580, 270270, 464100, 771290, 1245488, 1960610, 3016820, 4547840, 6729800, 9791859, 14028850, 19816225, 27627600, 38055225, 51833730, 69867525, 93262260, 123360780, 161784040, 210477476, 271763360

Offset: 0

N. J. A. Sloane

In the Carlitz et al. reference a(n)= Q_{5,n+2}(2), n >= 0, with a(n)=binomial(11+n,n+2)-(n+3)*binomial(n+6,n+2), (eq.(3.3), p. 356, with n=5, m->n+2,r=2). Q_{5,m}(2) is the number of sequences (i_1,i_2,...,i_m) with i_s, s=1,...,m, from {1,2,3,4,5} (repetitions allowed), with exactly 2 increases between successive elements (first position is counted as an increase).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374, p. 351.
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

CoefficientList[Series[(10-20*x+15*x^2-4*x^3)/(1-x)^10,{x,0,50}],x](* Vincenzo Librandi, Mar 28 2012 *)

a(n) = polcoeff((1+x+x^2+x^3+x^4)^(n+3), 9); \\ Joerg Arndt, Aug 04 2015

a(n) = A035343(n+3, 9) = binomial(n+6, 6)*(n^3+42*n^2+677*n+5040)/(9!/6!).
G.f.: (10-20*x+15*x^2-4*x^3)/(1-x)^10; numerator polynomial is N5(9, x) from the array A063422.
a(n) = 10*C(n+3,3) + 40*C(n+3,4) + 65*C(n+3,5) + 56*C(n+3,6) + 28*C(n+3,7) + 8*C(n+3,8) + C(n+3,9) (see comment in A213887). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = Sum_{k=1..10} (-1)^k * binomial(10,k) * a(n-k), a(0)=10. - G. C. Greubel, Aug 03 2015
a(n) = [x^9] (1+x+x^2+x^3+x^4)^(n+3). - Joerg Arndt, Aug 04 2015

Comments and more terms from Wolfdieter Lang, Aug 29 2001

More terms from Sean A. Irvine, Nov 24 2010

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).

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

A064056 Seventh column of quintinomial coefficients.

Original entry on oeis.org

3, 19, 68, 185, 426, 875, 1652, 2922, 4905, 7887, 12232, 18395, 26936, 38535, 54008, 74324, 100623, 134235, 176700, 229789, 295526, 376211, 474444, 593150, 735605, 905463, 1106784, 1344063, 1622260, 1946831

Offset: 0

Wolfdieter Lang, Aug 29 2001

Cf. A027659 (sixth column).

a(n) = A035343(n+2, 6) = binomial(n+2, 2)*(n^4+24*n^3+221*n^2+954*n+1080)/(6!/2!), n >= 0.
G.f.: (3-2*x-2*x^2+3*x^3-x^4)/(1-x)^7; numerator polynomial is N5(6, x) from the array A063422.
a(n) = 3*C(n+2,2) + 10*C(n+2,3) + 10*C(n+2,4) + 5*C(n+2,5) + C(n+2,6) (see comment in A213887). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A064057 Eighth column of quintinomial coefficients.

Original entry on oeis.org

2, 18, 80, 255, 666, 1520, 3144, 6030, 10890, 18722, 30888, 49205, 76050, 114480, 168368, 242556, 343026, 477090, 653600, 883179, 1178474, 1554432, 2028600, 2621450, 3356730, 4261842, 5368248

Offset: 0

Wolfdieter Lang, Aug 29 2001

Cf. A064056 (seventh column).

a(n) = A035343(n+2, 7)= binomial(n+3, 3)*(n+14)*(n^3+15*n^2+116*n+120)/(7!/3!).
G.f.: (2+2*x-8*x^2+7*x^3-2*x^4 )/(1-x)^8; numerator polynomial is N5(7, x) from the array A063422.
a(n) = 2*C(n+2,2) + 12*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 A213887). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A064058 Ninth column of quintinomial coefficients.

Original entry on oeis.org

1, 15, 85, 320, 951, 2415, 5475, 11385, 22110, 40612, 71214, 120055, 195650, 309570, 477258, 718998, 1061055, 1537005, 2189275, 3070914, 4247617, 5800025, 7826325, 10445175, 13798980, 18057546, 23422140

Offset: 0

Wolfdieter Lang, Aug 29 2001

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A064057 (eighth column), A000575 (tenth column).

With[{c=8!/4!},Table[(Binomial[n+4,4](n^4+34n^3+451n^2+2874n+1680))/c, {n,0,30}]] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,15,85,320,951,2415,5475,11385,22110},30] (* Harvey P. Dale, Oct 30 2011 *)

a(n) = A035343(n+2, 8) = binomial(n+4, 4)*(n^4+34*n^3+451*n^2+2874*n+1680)/(8!/4!).
G.f.: (1+6*x-14*x^2+11*x^3-3*x^4)/(1-x)^9; numerator polynomial is N5(8, x) from the array A063422.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) with a(0)=1, a(1)=15, a(2)=85, a(3)=320, a(4)=951, a(5)=2415, a(6)=5475, a(7)=11385, a(8)=22110. - Harvey P. Dale, Oct 30 2011
a(n) = C(n+2,2) + 12*C(n+2,3) + 31*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 A213887). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

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A213888 Triangle of coefficients of representations of columns of A213744 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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A000575 Tenth column of quintinomial coefficients.

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

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A064056 Seventh column of quintinomial coefficients.

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A064057 Eighth column of quintinomial coefficients.

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A064058 Ninth column of quintinomial coefficients.

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