A030528 -id:A030528 - cp's OEIS frontend

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

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).

A172383 a(0)=1, otherwise a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1,k)a(n-1-2k).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 19, 46, 118, 322, 903, 2653, 8053, 25194, 81387, 269667, 917529, 3197480, 11393821, 41497060, 154186653, 584151512, 2254240317, 8852998343, 35361762709, 143540660088, 591802631729, 2476701062087

Offset: 0

Paul Barry, Feb 01 2010

			Eigensequence for number triangle
  1;
  1,  0;
  0,  1,  0;
  1,  0,  1,  0;
  0,  2,  0,  1,  0;
  1,  0,  3,  0,  1,  0;
  0,  3,  0,  4,  0,  1,  0;
  1,  0,  6,  0,  5,  0,  1,  0;
  0,  4,  0, 10,  0,  6,  0,  1,  0;
  1,  0, 10,  0, 15,  0,  7,  0,  1,  0;
  0,  5,  0, 20,  0, 21,  0,  8,  0,  1,  0;
(augmented version of Riordan array (1/(1-x^2), x/(1-x^2)), A030528.

Seiichi Manyama, Table of n, a(n) for n = 0..932

Cf. A030528.

A172383 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        add(binomial(n-k-1,k)*procname(n-1-2*k),k=0..floor((n-1)/2)) ;
    end if;
end proc:
seq(A172383(n),n=0..20) ; # R. J. Mathar, Feb 11 2015

a[n_]:= If[n == 0, 1, Sum[Binomial[n-k-1, k]*a[n-2*k-1], {k, 0, Floor[(n-1)/2]}]]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Oct 07 2018 *)

G.f. A(x) satisfies: A(x) = 1 + (x/(1-x^2)) * A(x/(1-x^2)).

Name corrected by R. J. Mathar, Feb 11 2015

A307501 Expansion of Product_{k>=1} (1 + (x*(1 - x))^k).

Original entry on oeis.org

1, 1, 0, 0, -3, 1, -1, 3, 3, 0, -12, 15, -20, 5, 53, -113, 180, -241, 153, 173, -652, 787, 628, -4801, 11635, -18699, 20775, -12315, -6109, 21253, -7015, -61060, 174382, -260676, 190623, 130141, -549572, 399845, 1577502, -6670524, 14603574, -21111528, 16110192, 14794188, -82586174

Offset: 0

Ilya Gutkovskiy, Apr 11 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000009, A030528, A129519, A266108, A307496, A307500.

nmax = 44; CoefficientList[Series[Product[(1 + (x (1 - x))^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 44; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}] (x (1 - x))^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) Binomial[k, n - k] PartitionsQ[k], {k, 0, n}], {n, 0, 44}]

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d ) * (x*(1 - x))^k/k).

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,n-k)*A000009(k).

A049324 A convolution triangle of numbers generalizing Pascal's triangle A007318.

Original entry on oeis.org

1, 3, 1, 3, 6, 1, 0, 15, 9, 1, 0, 18, 36, 12, 1, 0, 9, 81, 66, 15, 1, 0, 0, 108, 216, 105, 18, 1, 0, 0, 81, 459, 450, 153, 21, 1, 0, 0, 27, 648, 1305, 810, 210, 24, 1, 0, 0, 0, 594, 2673, 2970, 1323, 276, 27, 1, 0, 0, 0, 324, 3915, 7938

Offset: 1

Wolfdieter Lang

			{1}; {3,1}; {3,6,1}; {0,15,9,1}; {0,18,36,12,1}; ...

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

a(n, m) := s1(-2, n, m), a member of a sequence of triangles including s1(0, n, m)= A023531(n, m) (unit matrix) and s1(2, n, m)=A007318(n-1, m-1) (Pascal's triangle). s1(-1, n, m)= A030528.

Cf. A049348, A049404.

a(n, m) = 3*(3*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, nA033842(2, m)).

A284938 Triangle read by rows: coefficients of the edge cover polynomial for the n-path graph P_n.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Apr 06 2017

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

Eric Weisstein's World of Mathematics, Edge Cover Polynomial
Eric Weisstein's World of Mathematics, Path Graph

Unsigned version of A057094.
Row sums are A000045(n-1).
Cf. A286912, A258993, A030528, A085478, A098925, A102426, A143858, etc.

Prepend[CoefficientList[Table[x^(n/2) Fibonacci[n - 1, Sqrt[x]], {n, 2, 14}], x], {0}] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
Prepend[CoefficientList[LinearRecurrence[{x, x}, {0, x}, {2, 14}], x], {0}] // Flatten (* Eric W. Weisstein, Apr 07 2017 *)

a(n) = abs(A057094(n)).

A339494 T(n, k) is the number of domino towers of n bricks with height at most 3 and k bricks in the base floor. Triangle read by rows, T(n, k) for 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 5, 9, 4, 1, 3, 14, 14, 5, 1, 1, 16, 29, 20, 6, 1, 0, 12, 46, 51, 27, 7, 1, 0, 5, 52, 101, 81, 35, 8, 1, 0, 1, 41, 150, 190, 120, 44, 9, 1, 0, 0, 22, 169, 345, 323, 169, 54, 10, 1, 0, 0, 7, 143, 495, 687, 511, 229, 65, 11, 1

Offset: 1

Peter Luschny, Dec 07 2020

This is the third triangle in a sequence of triangles: The first is the unit triangle A023531; the second is the binomial triangle C(k, n-k) without the first column, triangle A030528. This triangle highlights the connection between the Pascal triangle and the Fibonacci numbers in the case m = 2. Similarly, the current triangle and its row sums generalizes this to the case m = 3 of the construction of Union(A333650(n, j), j=1..m), classified by the number of bricks in the base floor.

			Triangle starts:        n: [row] sum
                          1: [1] 1
                         2: [2, 1] 3
                       3: [5, 3, 1] 9
                     4: [5, 9, 4, 1] 19
                   5: [3, 14, 14, 5, 1] 37
                 6: [1, 16, 29, 20, 6, 1] 73
              7: [0, 12, 46, 51, 27, 7, 1] 144
            8: [0, 5, 52, 101, 81, 35, 8, 1] 283
         9: [0, 1, 41, 150, 190, 120, 44, 9, 1] 556
     10: [0, 0, 22, 169, 345, 323, 169, 54, 10, 1] 1093

Peter Luschny, Domino towers classified by the size of the base floor. Illustrating row 4 of the triangle.

Cf. A339495 (row sums), A333650, A030528, A023531.

A005119 Infinitesimal generator of x*(x + 1).

Original entry on oeis.org

1, 1, 3, 16, 124, 1256, 15576, 226248, 3729216, 68179968, 1361836800, 29501349120, 693638208000, 17815908096000, 502048890201600, 15388268595840000, 500579319427891200, 16817771937344716800, 581609175119297740800

Offset: 1

N. J. A. Sloane, Simon Plouffe

From Peter Bala, Dec 09 2015: (Start)
Given a formal power series f(x) = x + f_2*x^2 + f_3*x^3 + ... Labelle [Section 4, Proposition 4] shows there is a power series w(x) = w_2*x^2 + w_3*x^3 + w_4*x^4 + ..., called the infinitesimal generator of f, such that the n-fold composition f^(n)(x) = f o f o ... o f (n factors) of f(x) is given by the operator exp( n*w(x)*d/dx ) acting on x. This gives the expansion f^(n)(x) = x + n/1!*w(x) + n^2/2!*w(x)*w'(x) + .... Taking n = -1 gives an expansion for the series reversion of f(x).
Let R denote the Riordan array (f(x)/x, f(x)). Then the coefficients of the infinitesimal generator w(x) form the first column of the matrix logarithm log(R).
Here we take f(x) = x + x^2 and calculate w(x) = x^2*(1 - x + 3*x^2/2! - 16*x^3/3! + 124*x^4/4! - ...). The numerators of the coefficients give a signed version of the present sequence. See the example below. (End)
a(29) = -307081193389527408920486163460915200000 is the first negative term. Georg Fischer, Feb 15 2019

			From _Peter Bala_, Dec 09 2015: (Start)
The Riordan array R = (1 + x, x*(1 + x)) is A030528.
log(R) begins
  /    0
  |    1          0
  |   -1         1*2        0
  |  3/2!       -1*2       1*3    0
  |-16/3!   (3/2!)*2      -1*3   1*4   0
  |124/4! (-16/3!)*2  (3/2!)*3  -1*4  1*5  0
  |...
  \
The first column begins [1, -1, 3/2!, -16/3! 124/4!, ...]. (End)

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

G. C. Greubel, Table of n, a(n) for n = 1..200
Loïc Foissy, Cointeraction on noncrossing partitions and related polynomial invariants, arXiv:2501.18212 [math.CO], 2025. See pp. 21, 24.
Gilbert Labelle, Sur l'Inversion et l'Iteration Continue des Séries Formelles, European Journal of Combinatorics, Vol. 1 Issue 2 (June 1980), 113-138.

Cf. A030528.

max = 19; f[x_] := Sum[a[n+1]*x^n/n!, {n, 0, max}]; coes = CoefficientList[ Series[ f[x]-((1-x)^2/(1-2*x))*f[x-x^2], {x, 0, max}], x]; Array[a, max] /. Solve[a[1] == a[2] == 1 && Thread[coes == 0]][[1]] (* Jean-François Alcover, Nov 03 2011 *)
nmax=20; a = ConstantArray[0,nmax]; a[[1]]=1; Do[a[[n]] = (n-2)! *Sum[(-1)^(i+1)*Binomial[n-i+1,i+1]*a[[n-i]]/(n-i-1)!,{i,1,n-1}],{n,2,nmax}]; a (* Vaclav Kotesovec, Mar 12 2014 *)

{a(n)=if(n<1,0,if(n==1,1,(n-2)!*sum(i=1,n-1,(-1)^(i+1)*binomial(n-i+1,i+1)*a(n-i)/(n-i-1)!)))} \\ Paul D. Hanna, Dec 27 2007

a(n) = (n-2)!*Sum_{i=1..n-1} (-1)^(i+1)*C(n-i+1,i+1)*a(n-i)/(n-i-1)! for n>1 with a(1)=1. E.g.f. satisfies: A(x) = (1-x)^2/(1-2x)*A(x-x^2) where A(x) = Sum_{n>=0}a(n+1)*x^n/n! with offset so that A(0)=1. - Paul D. Hanna, Dec 27 2007

More terms from Paul D. Hanna, Dec 27 2007

A049325 A convolution triangle of numbers generalizing Pascal's triangle A007318.

Original entry on oeis.org

1, 6, 1, 16, 12, 1, 16, 68, 18, 1, 0, 224, 156, 24, 1, 0, 448, 840, 280, 30, 1, 0, 512, 3072, 2080, 440, 36, 1, 0, 256, 7872, 10896, 4160, 636, 42, 1, 0, 0, 14080, 42240, 28240, 7296, 868, 48, 1, 0, 0, 16896, 123904, 145376, 60720, 11704, 1136, 54, 1, 0, 0, 12288

Offset: 1

Wolfdieter Lang

			{1}; {6,1}; {16,12,1}; {16,68,18,1}; {0,224,156,24,1}; ...

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

a(n, m) := s1(-3, n, m), a member of a sequence of triangles including s1(0, n, m)= A023531(n, m) (unit matrix) and s1(2, n, m)=A007318(n-1, m-1) (Pascal's triangle). s1(-1, n, m)= A030528, s1(-2, n, m)= A049324(n, m).

Cf. A049349.

a(n, m) = 4*(4*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, nA033842(3, m)).

A049326 A convolution triangle of numbers generalizing Pascal's triangle A007318.

Original entry on oeis.org

1, 10, 1, 50, 20, 1, 125, 200, 30, 1, 125, 1250, 450, 40, 1, 0, 5250, 4375, 800, 50, 1, 0, 15000, 30375, 10500, 1250, 60, 1, 0, 28125, 157500, 100500, 20625, 1800, 70, 1, 0, 31250, 621875, 740000, 250625, 35750, 2450, 80, 1, 0, 15625, 1875000, 4318750

Offset: 1

Wolfdieter Lang

			{1}; {10,1}; {50,20,1}; {125,200,30,1}; {125,1250,450,40,1}; ...

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

a(n, m) := s1(-4, n, m), a member of a sequence of triangles including s1(0, n, m)= A023531(n, m) (unit matrix) and s1(2, n, m)=A007318(n-1, m-1) (Pascal's triangle). s1(-1, n, m)= A030528.

Cf. A049350.

a(n, m) = 5*(5*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, nA033842(4, m)).

cp's OEIS Frontend

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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A172383 a(0)=1, otherwise a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1,k)*a(n-1-2*k).

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A307501 Expansion of Product_{k>=1} (1 + (x*(1 - x))^k).

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A049324 A convolution triangle of numbers generalizing Pascal's triangle A007318.

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A284938 Triangle read by rows: coefficients of the edge cover polynomial for the n-path graph P_n.

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A339494 T(n, k) is the number of domino towers of n bricks with height at most 3 and k bricks in the base floor. Triangle read by rows, T(n, k) for 1 <= k <= n.

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A005119 Infinitesimal generator of x*(x + 1).

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A172383 a(0)=1, otherwise a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1,k)a(n-1-2k).