A098569 -id:A098569 - cp's OEIS frontend

A098568 Triangle of triangular binomial coefficients, read by rows, where column k has the g.f.: 1/(1-x)^((k+1)*(k+2)/2) for k >= 0.

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 21, 10, 1, 1, 15, 56, 55, 15, 1, 1, 21, 126, 220, 120, 21, 1, 1, 28, 252, 715, 680, 231, 28, 1, 1, 36, 462, 2002, 3060, 1771, 406, 36, 1, 1, 45, 792, 5005, 11628, 10626, 4060, 666, 45, 1, 1, 55, 1287, 11440, 38760, 53130, 31465, 8436

Offset: 0

Paul D. Hanna, Sep 15 2004

The row sums form A098569: {1,2,5,14,43,143,510,1936,7775,32869,...}. How do the terms of row k tend to be distributed as k grows?
Remarkably, column k of the matrix inverse (A121434) equals signed column k of the triangular matrix power: A107876^(k*(k+1)/2) for k >= 0. - Paul D. Hanna, Aug 25 2006
Surprisingly, the row sums (A098569) equal the row sums of triangle A131338. - Paul D. Hanna, Aug 30 2007
Number of sequences S = s(1)s(2)...s(n) such that S contains m 0's, for 1<=j<=n, s(j)Frank Ruskey, Apr 15 2011
As a rectangular array read by antidiagonals R(n,k) (n>=2, k>=0) is the number of labeled graphs on n nodes that have exactly k arcs where multiple arcs are allowed to connect distinct vertex pairs. R(n,k) = C(C(n,2)+k-1,k). See example below. - Geoffrey Critzer, Nov 12 2011

			G.f.s of columns: 1/(1-x), 1/(1-x)^3, 1/(1-x)^6, 1/(1-x)^10, 1/(1-x)^15, ...
Rows begin:
  1;
  1,  1;
  1,  3,    1;
  1,  6,    6,     1;
  1, 10,   21,    10,      1;
  1, 15,   56,    55,     15,      1;
  1, 21,  126,   220,    120,     21,      1;
  1, 28,  252,   715,    680,    231,     28,     1;
  1, 36,  462,  2002,   3060,   1771,    406,    36,     1;
  1, 45,  792,  5005,  11628,  10626,   4060,   666,    45,    1;
  1, 55, 1287, 11440,  38760,  53130,  31465,  8436,  1035,   55,  1;
  1, 66, 2002, 24310, 116280, 230230, 201376, 82251, 16215, 1540, 66, 1; ...
From _Frank Ruskey_, Apr 15 2011: (Start)
In reference to comment about s(1)s(2)...s(n) above,
   a(4,2) = 6 = |{0012, 0013, 0023, 0101, 0103, 0120}|  and
   a(4,3) = 6 = |{0001, 0002, 0003, 0010, 0020, 0100}|. (End)
From _Geoffrey Critzer_, Nov 12 2011: (Start)
In reference to comment about multigraphs above,
  1,    1,    1,    1,    1,     1,     ...  2 nodes
  1,    3,    6,    10,   15,    21,    ...  3 nodes
  1,    6,    21,   56,   126,   252,   ...  .
  1,    10,   55,   220,  715,   2002,  ...  .
  1,    15,   120,  680,  3060,  11628, ...  .
  1,    21,   231,  1771, 10626, 58130, ...  . (End)

Paul D. Hanna, Table of n, a(n) for n = 0..1080, of flattened triangle, read by rows 0..45.
Soheir M. Khamis, Height counting of unlabeled interval and N-free posets, Discrete Math. 275 (2004), no. 1-3, 165-175.
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
Zhicong Lin and Shishuo Fu, On 120-avoiding inversion and ascent sequences, arXiv:2003.11813 [math.CO], 2020.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], (2017), table 60.

Cf. A098569. A290428 (unlabeled graphs).
Cf. A121434 (inverse); variants: A122175, A122176, A122177; A107876.
Cf. A131338.

t[n_, k_] = Binomial[(k+1)*(k+2)/2 + n-k-1, n-k]; Flatten[Table[t[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, Jul 18 2011 *)

{T(n,k)=binomial((k+1)*(k+2)/2+n-k-1,n-k)}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

T(n, k) = binomial((k+1)*(k+2)/2 + n-k-1, n-k).

A131338 Triangle, read by rows of n*(n+1)/2 + 1 terms, that starts with a '1' in row 0 with row n consisting of n '1's followed by the partial sums of the prior row.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 3, 4, 6, 9, 14, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 20, 29, 43, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 27, 37, 51, 71, 100, 143, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 35, 46, 61, 81, 108, 145, 196

Offset: 0

Paul D. Hanna, Jun 29 2007

			Triangle begins:
1;
1, 1;
1,1, 1,2;
1,1,1, 1,2,3,5;
1,1,1,1, 1,2,3,4,6,9,14;
1,1,1,1,1, 1,2,3,4,5,7,10,14,20,29,43;
1,1,1,1,1,1, 1,2,3,4,5,6,8,11,15,20,27,37,51,71,100,143;
1,1,1,1,1,1,1, 1,2,3,4,5,6,7,9,12,16,21,27,35,46,61,81,108,145,196,267,367,510; ...
Row sums equal the row sums (A098569) of triangle A098568,
where A098568(n, k) = binomial( (k+1)*(k+2)/2 + n-k-1, n-k):
1;
1, 1;
1, 3, 1;
1, 6, 6, 1;
1, 10, 21, 10, 1;
1, 15, 56, 55, 15, 1;
1, 21, 126, 220, 120, 21, 1; ...

Paul D. Hanna, Rows n = 0..16, flattened.

Cf. A098568, A098569 (row sums), A121690, A183202.

Cf. A214403 (variant).

T(n,k)=if(k>n*(n+1)/2 || k<0,0,if(k<=n,1,sum(i=0,k-n,T(n-1,i))))
for(n=0, 10, for(k=0, n*(n+1)/2, print1(T(n, k), ", ")); print(""))

T(n,k) = Sum_{i=0..k-n} T(n-1,i) for k>n, else T(n,k)=1 for n>=k>=0.
Right border: T(n+1, (n+1)*(n+2)/2) = A098569(n) = Sum_{k=0..n} C( (k+1)*(k+2)/2 + n-k-1, n-k).
T(n, n*(n-1)/2 + 1) = Sum_{k=0..n-1} C(k*(k+1)/2, n-k) = A121690(n-1) for n>=1. - Paul D. Hanna, Aug 30 2007

A326423 G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n(n+1)/2) x^n = Sum_{n>=0} x^n / (1-x)^(n*(n-1)/2).

Original entry on oeis.org

1, 0, 1, 1, 4, 11, 39, 147, 598, 2577, 11669, 55156, 270938, 1378577, 7247494, 39290662, 219304105, 1258592815, 7418414658, 44863100701, 278117328554, 1765909629266, 11475651209600, 76267987517000, 518046275820877, 3593989140928928, 25450794447346211, 183860936257142088, 1354254148649619126, 10164913983190913353, 77710718331267769117

Offset: 0

Paul D. Hanna, Jul 03 2019

nonn

			G.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 11*x^5 + 39*x^6 + 147*x^7 + 598*x^8 + 2577*x^9 + 11669*x^10 + 55156*x^11 + 270938*x^12 + 1378577*x^13 + 7247494*x^14 + ...
such that the following series are equal
B(x) = 1 + A(x)*x + A(x)^3*x^2 + A(x)^6*x^3 + A(x)^10*x^4 + A(x)^15*x^5 + A(x)^21*x^6 + A(x)^28*x^7 + A(x)^36*x^8 + A(x)^45*x^9 + ...
and
B(x) = 1 + x + x^2/(1-x) + x^3/(1-x)^3 + x^4/(1-x)^6 + x^5/(1-x)^10 + x^6/(1-x)^15 + x^7/(1-x)^21 + x^8/(1-x)^28 + x^9/(1-x)^36 + ...
where
B(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 43*x^6 + 143*x^7 + 510*x^8 + 1936*x^9 + 7775*x^10 + 32869*x^11 + 145665*x^12 + ... + A098569(n-1)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A098569, A326424, A325294.

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A]=polcoeff( sum(m=0,#A, x^m/(1-x +x*O(x^#A))^(m*(m-1)/2) - x^m*Ser(A)^(m*(m+1)/2) ),#A)); A[n+1]}
for(n=0,35,print1(a(n),", "))

A183202 Triangle, read by rows, where T(n,k) equals the sum of (n-k) terms in row n of triangle A131338 starting at position nk - k(k-1)/2, with the main diagonal formed from the row sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 3, 3, 5, 4, 6, 10, 9, 14, 5, 10, 22, 34, 29, 43, 6, 15, 40, 84, 122, 100, 143, 7, 21, 65, 169, 334, 463, 367, 510, 8, 28, 98, 300, 738, 1390, 1851, 1426, 1936, 9, 36, 140, 489, 1426, 3345, 6043, 7767, 5839, 7775, 10, 45, 192, 749, 2510, 6990, 15735

Offset: 0

Paul D. Hanna, Dec 30 2010

			Triangle begins:
1;
1,1;
2,1,2;
3,3,3,5;
4,6,10,9,14;
5,10,22,34,29,43;
6,15,40,84,122,100,143;
7,21,65,169,334,463,367,510;
8,28,98,300,738,1390,1851,1426,1936;
9,36,140,489,1426,3345,6043,7767,5839,7775;
10,45,192,749,2510,6990,15735,27374,34097,25094,32869; ...
The rows are derived from triangle A131338 by summing terms in the following manner:
(1);
(1),(1);
(1+1),(1),(2);
(1+1+1),(1+2),(3),(5);
(1+1+1+1),(1+2+3),(4+6),(9),(14);
(1+1+1+1+1),(1+2+3+4),(5+7+10),(14+20),(29),(43);
(1+1+1+1+1+1),(1+2+3+4+5),(6+8+11+15),(20+27+37),(51+71),(100),(143); ...
where row n of triangle A131338 consists of n '1's followed by the partial sums of the prior row.

Cf. A131338, A098568, A098569 (row sums), A183203 (antidiagonal sums).

{A131338(n, k)=if(k>n*(n+1)/2||k<0,0,if(k<=n,1,sum(i=0, k-n,A131338(n-1,i))))}
{T(n,k)=if(n==k,A131338(n,n*(n+1)/2),sum(j=n*k-k*(k-1)/2,n*k-k*(k-1)/2+n-k-1,A131338(n,j)))}

cp's OEIS Frontend

A098568 Triangle of triangular binomial coefficients, read by rows, where column k has the g.f.: 1/(1-x)^((k+1)*(k+2)/2) for k >= 0.

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A131338 Triangle, read by rows of n*(n+1)/2 + 1 terms, that starts with a '1' in row 0 with row n consisting of n '1's followed by the partial sums of the prior row.

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A326423 G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n(n+1)/2) x^n = Sum_{n>=0} x^n / (1-x)^(n*(n-1)/2).

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A183202 Triangle, read by rows, where T(n,k) equals the sum of (n-k) terms in row n of triangle A131338 starting at position nk - k(k-1)/2, with the main diagonal formed from the row sums.

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cp's OEIS Frontend

A098568 Triangle of triangular binomial coefficients, read by rows, where column k has the g.f.: 1/(1-x)^((k+1)*(k+2)/2) for k >= 0.

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Mathematica

PARI

Formula

A131338 Triangle, read by rows of n*(n+1)/2 + 1 terms, that starts with a '1' in row 0 with row n consisting of n '1's followed by the partial sums of the prior row.

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A326423 G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n*(n+1)/2) * x^n = Sum_{n>=0} x^n / (1-x)^(n*(n-1)/2).

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A183202 Triangle, read by rows, where T(n,k) equals the sum of (n-k) terms in row n of triangle A131338 starting at position nk - k(k-1)/2, with the main diagonal formed from the row sums.

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A326423 G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n(n+1)/2) x^n = Sum_{n>=0} x^n / (1-x)^(n*(n-1)/2).