A102086 -id:A102086 - cp's OEIS frontend

A102087 Column 1 of triangular matrix A102086, which shifts upward to exclude the main diagonal under matrix square.

0, 2, 4, 20, 156, 1664, 22684, 378572, 7504640, 172785512, 4540756252, 134330010172, 4423176368332, 160596613105384, 6378859853838480, 275308217428662672, 12836003750434047344, 643227594173121801096

Offset: 0

Paul D. Hanna, Dec 29 2004

nonn

Cf. A102086, A102088.

T[n_, n_] := n+1; T[n_, k_] /; k>n = 0; T[n_, k_] /; k == n-1 := n^2; T[n_, k_] := T[n, k] = Coefficient[1-Sum[T[i, k]*x^i*Product[1-(j+k)*x, {j, 1, i-k+1}], {i, k, n-1}], x, n]; a[n_] := T[n, 1]; Table[a[n], {n, 0, 17} ] (* Jean-François Alcover, Dec 15 2014 *)

{a(n)=local(A=matrix(2,2),B);A[1,1]=1; for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=j,if(j==1,B[i,j]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]));));A=B); return(A[n+1,2])}

A102088 Row sums of triangular matrix A102086, which shifts upward to exclude the main diagonal under matrix square.

Original entry on oeis.org

1, 3, 10, 49, 367, 3850, 52173, 868614, 17199370, 395757887, 10396896795, 307511681367, 10124396776169, 367567456615441, 14598938725992903, 630060602243145513, 29375322688053255480, 1472008290120323375502

Offset: 0

Paul D. Hanna, Dec 29 2004

nonn

Cf. A102086, A102087.

{a(n)=local(A=matrix(2,2),B);A[1,1]=1; for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=j,if(j==1,B[i,j]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]));));A=B); return(sum(k=0,n,A[n+1,k+1]))}

A082161 Number of deterministic completely defined initially connected acyclic automata with 2 inputs and n transient unlabeled states (and a unique absorbing state).

Original entry on oeis.org

1, 3, 16, 127, 1363, 18628, 311250, 6173791, 142190703, 3737431895, 110577492346, 3641313700916, 132214630355700, 5251687490704524, 226664506308709858, 10568175957745041423, 529589006347242691143, 28395998790096299447521

Offset: 1

Valery A. Liskovets, Apr 09 2003

Coefficients T_2(n,k) form the array A082169. These automata have no nontrivial automorphisms (by states).

Also counts the relaxed compacted binary trees of size n. A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. See the Genitrini et al. link. - Michael Wallner, Apr 20 2017

			a(2)=3 since the following transition diagrams represent all three initially connected acyclic automata with two input letters x and y, two transient states 1 (initial) and 2 and the absorbing state 0:
  1 == x, y==> 2 == x, y ==> 0 == x, y ==> 0, 1 -- x --> 2 == x, y ==> 0 == x, y ==> 0
  1 -- y --> 0
and the last one with x and y interchanged.

Roland Bacher and Christophe Reutenauer, The number of right ideals of given codimension over a finite field, in Noncommutative Birational Geometry, Representations and Combinatorics, edited by Arkady. Berenstein and Vladimir. Retakha, Contemporary Mathematics, Vol. 592, 2013.

Vaclav Kotesovec, Table of n, a(n) for n = 1..350
David Callan, A determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata, arXiv:0704.0004 [math.CO], 2007.
Manosij Ghosh Dastidar and Michael Wallner, Asymptotics of relaxed k-ary trees, arXiv:2404.08415 [math.CO], 2024. See p. 1.4.
Andrew Elvey Price, Wenjie Fang, and Michael Wallner, Compacted binary trees admit a stretched exponential, arXiv:1908.11181 [math.CO], 2019-2020; J. Combin. Theory Ser. A 177 (2021), Paper No. 105306, 40 pp.
Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic enumeration of compacted binary trees of bounded right height, arXiv:1703.10031 [math.CO], 2017; J. Combin. Theory Ser. A 172 (2020), 105177, 49 pp.
Valery A. Liskovets, Exact enumeration of acyclic automata, Proc. 15th Conf. "Formal Power Series and Algebr. Combin. (FPSAC'03)", 2003.
Valery A. Liskovets, Exact enumeration of acyclic deterministic automata, Discrete Appl. Math., 154, No.3 (2006), 537-551.
Fedor Petrov, On a generating function and vector ν of length n, answer to question on MathOverflow (2024).
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017-2018; Theoret. Comput. Sci. 755 (2019), 1-12.

Cf. A008276, A082157, A082169, A102086, A102316, A254789.

a[n_]:= a[n]= If[n==0, 1, Coefficient[1-Sum[a[k]*x^k*Product[1-j*x, {j, 1, k+1}], {k, 0, n-1}], x, n]];
Table[a[n], {n, 18}] (* Jean-François Alcover, Dec 15 2014, after Paul D. Hanna *)

{a(n)=if(n==0,1,polcoeff(1-sum(k=0,n-1,a(k)*x^k*prod(j=1,k+1,1-j*x+x*O(x^n))),n))} \\ Paul D. Hanna, Jan 07 2005

{a(n)=local(A);if(n<1,0,A=x+x*O(x^n); for(k=0,n,A+=polcoeff(A,k)*x^k*(1-prod(i=1,k+1,1-i*x))); polcoeff(A,n))} /* Michael Somos, Jan 16 2005 */

upto(n) = my(v=vector(n+1, i, i==1)); for(i=1, n, for(j=i+1, n+1, v[j] += i*v[j-1])); v[2..#v] \\ Mikhail Kurkov, Oct 25 2024

from functools import cache
@cache
def b(n, k):
    if n == 0: return k + 1
    return sum(b(j, k)*b(n-j-1, k+j) for j in range(n))
def A082161(n): return b(n, 0)
print([A082161(n) for n in range(1, 19)]) # G. C. Greubel, Jan 18 2024

a(n) = c_2(n)/(n-1)! where c_2(n) = T_2(n, 1) - Sum_{j=1..n-1} binomial(n-1, j-1)*T_2(n-j, j+1)*c_2(j), and T_2(0, k) = 1, T_2(n, k) = Sum_{i=0..n-1} binomial(n, i)*(-1)^(n-i-1)*(i+k)^(2*n-2*i)*T_2(i, k), n > 0.
Equals column 0 of triangle A102086. Also equals main diagonal of A102316: a(n) = A102086(n, 0) = A102316(n, n). - Paul D. Hanna, Jan 07 2005
G.f.: 1 = Sum_{n>=0} a(n)*x^n*prod_{k=1, n+1} (1-k*x) for n>0 with a(0)=1. a(n) = -Sum_{k=1, [(n+1)/2]} A008276(n-k+1, k)*a(n-k) where A008276 is Stirling numbers of the first kind. Thus G.f.: 1 = (1-x) + 1*x*(1-x)(1-2x) + 3*x^2*(1-x)(1-2x)(1-3x) + ... + a(n)*x^n*(1-x)(1-2x)(1-3x)*..*(1-(n+1)*x) + ... with a(0)=1. - Paul D. Hanna, Jan 14 2005
a(n) is the determinant of the n X n matrix with (i, j) entry = StirlingCycle[i+1, 2i-j]. - David Callan, Jul 20 2005
a(n) = b(n,0) where b(0,p) = p+1 and b(n+1,p) = Sum_{i=0..n} b(i,p)*b(n-i,p+i) for n>=1. - Michael Wallner, Apr 20 2017
From Michael Wallner, Jan 31 2022: (Start)
a(n) = r(n,n) where r(n,m)=(m+1)*r(n-1,m)+r(n,m-1) for n>=m>=1, r(n,m)=0 for n=0.
a(n) = Theta(n!*4^n*exp(3*a1*n^(1/3))*n) for large n, where a1=-2.338... is the largest root of the Airy function Ai(x) of the first kind; see [Elvey Price, Fang, Wallner 2021]. (End)

A102098 Triangular matrix, read by rows, that satisfies: T(n,k) = [T^3](n-1,k) when n>k>=0, with T(n,n) = (n+1).

Original entry on oeis.org

1, 1, 2, 7, 8, 3, 139, 152, 27, 4, 5711, 6200, 999, 64, 5, 408354, 442552, 69687, 3904, 125, 6, 45605881, 49399320, 7724835, 416704, 11375, 216, 7, 7390305396, 8003532512, 1248465852, 66464960, 1725875, 27432, 343, 8, 1647470410551

Offset: 0

Paul D. Hanna, Dec 29 2004

Column 0 forms A082162. Column 1 forms A102099. Row sums form A102100. This triangle is a variant of A102086.

			Rows of T begin:
[1],
[1,2],
[7,8,3],
[139,152,27,4],
[5711,6200,999,64,5],
[408354,442552,69687,3904,125,6],
[45605881,49399320,7724835,416704,11375,216,7],
[7390305396,8003532512,1248465852,66464960,1725875,27432,343,8],...
Matrix cube T^3 equals T excluding the main diagonal:
[1],
[7,8],
[139,152,27],
[5711,6200,999,64],
[408354,442552,69687,3904,125],...

Cf. A082162, A102099, A102100, A102086.

{T(n,k)=local(A=matrix(1,1),B);A[1,1]=1; for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=j,if(j==1,B[i,j]=(A^3)[i-1,1], B[i,j]=(A^3)[i-1,j]));));A=B);return(A[n+1,k+1])}

T(n, 0) = A082162(n) for n>0, with T(0, 0) = 1.

A102316 Triangle, read by rows, where T(n,k) = T(n,k-1) + (k+1)*T(n-1,k) for n>k>0, T(n,0)=1 and T(n,n) = T(n,n-1) for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 7, 16, 16, 1, 15, 63, 127, 127, 1, 31, 220, 728, 1363, 1363, 1, 63, 723, 3635, 10450, 18628, 18628, 1, 127, 2296, 16836, 69086, 180854, 311250, 311250, 1, 255, 7143, 74487, 419917, 1505041, 3683791, 6173791, 6173791, 1, 511, 21940

Offset: 0

Paul D. Hanna, Jan 04 2005

Main diagonal is A082161 (with offset). Row sums give A102317.

T(n,k) = number of column-marked subdiagonal paths of steps east (1,0) and north (0,1) from the origin to (n,k). Subdiagonal means that the path never rises above the diagonal line y=x and column-marked means that for 1 <= i <= n, one unit square directly below the i-th east step and above the line y=-1 is marked. - David Callan, Feb 04 2006

			T(5,2) = 220 = 1*1 + 2*15 + 3*63 = 1*T(4,0) + 2*T(4,1) + 3*T(4,2).
T(5,2) = 220 = 31 + 3*63 = T(5,1) + (2+1)*T(4,2).
T(5,3) = 728 = 220 + 4*127 = T(5,2) + (3+1)*T(4,3).
Rows begin:
[1],
[1,1],
[1,3,3],
[1,7,16,16],
[1,15,63,127,127],
[1,31,220,728,1363,1363],
[1,63,723,3635,10450,18628,18628],
[1,127,2296,16836,69086,180854,311250,311250],
[1,255,7143,74487,419917,1505041,3683791,6173791,6173791],...

Cf. A102086, A082161, A102317.

PARI
```
T(n,k)=if(n
				
```

T(n, k) = Sum_{j=0..k} (j+1)*T(n-1, j) for n>k>0, T(n, 0)=1 for n>=0. T(n, n) = A082161(n) for n>0; A082161(n+1) = Sum_{k=0..n} (k+1)*T(n, k).

A102916 Triangle, read by rows, where the antidiagonals are formed by interleaving the rows of triangle A102098 with the rows of its matrix square (A102920).

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 3, 8, 9, 4, 7, 40, 27, 16, 5, 36, 152, 189, 64, 25, 6, 139, 1128, 999, 576, 125, 36, 7, 1036, 6200, 9720, 3904, 1375, 216, 49, 8, 5711, 61120, 69687, 47040, 11375, 2808, 343, 64, 9, 56355, 442552, 857466, 416704, 163500, 27432, 5145, 512

Offset: 0

Paul D. Hanna, Jan 21 2005

Column 0 is A102917, the interleaving of A082162 with A102921. Under matrix cube, triangle A102098 shifts each column up 1 row.

			Rows begin:
[1],
[1,2],
[1,4,3],
[3,8,9,4],
[7,40,27,16,5],
[36,152,189,64,25,6],
[139,1128,999,576,125,36,7],
[1036,6200,9720,3904,1375,216,49,8],
[5711,61120,69687,47040,11375,2808,343,64,9],...
The antidiagonals are formed by interleaving the
rows of triangle A102098:
[1],
[1,2],
[7,8,3],
[139,152,27,4],...
with the rows of the matrix square of A102098,
which is triangle A102920:
[1],
[3,4],
[36,40,9],
[1036,1128,189,16],...
G.f. for Column 0 (A102917): 1 = 1*(1-x) + 1*x*(1-x)
+ 1*x^2*(1-x)(1-2x) + 3*x^3*(1-x)(1-2x)
+ 7*x^4*(1-x)(1-2x)(1-3x) + 36*x^5*(1-x)(1-2x)(1-3x) +...
+ A082162(n)*x^(2n)*(1-x)(1-2x)*..*(1-(n+1)x)
+ A102921(n)*x^(2n+1)*(1-x)(1-2x)*..*(1-(n+1)x) + ...
G.f. for Column 1 (A102918): 2 = 2*(1-2x) + 4*x*(1-2x)
+ 8*x^2*(1-2x)(1-3x) + 40*x^3*(1-2x)(1-3x)
+ 152*x^4*(1-2x)(1-3x)(1-4x) + 1128*x^5*(1-2x)(1-3x)(1-4x) +...
+ T(2n+1,1)*x^(2n)*(1-2x)(1-3x)*..*(1-(n+2)x)
+ T(2n+2,1)*x^(2n+1)*(1-2x)(1-3x)*..*(1-(n+2)x) + ...

Cf. A102086, A102098, A102920, A102917, A102918.

PARI
```
{T(n,k)=if(n
				
```

G.f. for column k: T(k, k) = k+1 = Sum_{n>=0} T(n+k, k)*x^n*Product_{j=k..[n/2+k]} (1-(j+1)*x).

A102101 Triangular matrix, read by rows, that satisfies: T(n,k) = [T^4](n-1,k) when n>k>=0, with T(n,n) = (n+1).

Original entry on oeis.org

1, 1, 2, 15, 16, 3, 1000, 1040, 81, 4, 189035, 196080, 14175, 256, 5, 79278446, 82196224, 5866992, 94464, 625, 6, 63263422646, 65585046960, 4667640795, 73281280, 419375, 1296, 7, 86493299281972, 89664824687968, 6376139907030

Offset: 0

Paul D. Hanna, Dec 29 2004

Column 0 forms A102102. Column 1 forms A102103. Row sums form A102104. This triangle is a variant of A102086 and A102098.

			Rows of T begin:
[1],
[1,2],
[15,16,3],
[1000,1040,81,4],
[189035,196080,14175,256,5],
[79278446,82196224,5866992,94464,625,6],
[63263422646,65585046960,4667640795,73281280,419375,1296,7].
Matrix fourth power T^4 equals T excluding the main diagonal:
[1],
[15,16],
[1000,1040,81],
[189035,196080,14175,256],
[79278446,82196224,5866992,94464,625],...

Cf. A102102, A102103, A102104, A102086, A102098.

{T(n,k)=local(A=matrix(1,1),B);A[1,1]=1; for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=j,if(j==1,B[i,j]=(A^4)[i-1,1], B[i,j]=(A^4)[i-1,j]));));A=B);return(A[n+1,k+1])}

T(n, 0) = A082162(n) for n>0, with T(0, 0) = 1.

A102320 Triangular matrix, read by rows, that satisfies: T(n,k) = [T^2](n-1,k) when n>k>=0, with T(n,n) = (2*n+1).

Original entry on oeis.org

1, 1, 3, 4, 9, 5, 33, 72, 25, 7, 436, 945, 300, 49, 9, 8122, 17568, 5425, 784, 81, 11, 197920, 427770, 130700, 18081, 1620, 121, 13, 6007205, 12979080, 3947050, 535864, 45441, 2904, 169, 15, 219413116, 473981445, 143812400, 19348042, 1599588, 95953

Offset: 0

Paul D. Hanna, Jan 05 2005

Column 0 forms A102321. Column 1 forms A102322. The contribution of each term along the main diagonal to column 0 is given by triangle of coefficients A102323.

			Rows of T begin:
[1],
[1,3],
[4,9,5],
[33,72,25,7],
[436,945,300,49,9],
[8122,17568,5425,784,81,11],
[197920,427770,130700,18081,1620,121,13],
[6007205,12979080,3947050,535864,45441,2904,169,15],...
Matrix square T^2 equals T excluding the main diagonal:
[1],
[4,9],
[33,72,25],
[436,945,300,49],
[8122,17568,5425,784,81],...

Cf. A102086, A102321, A102322, A102323.

{T(n,k)=local(A=Mat(1),B); for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=2*j-1,if(j==1,B[i,j]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]));));A=B);return(A[n+1,k+1])}

A106208 Triangular matrix T, read by rows, that satisfies: [T^-1](n,k) = -(k+1)*T(n-1,k) when (n-1)>=k>=0, with T(n,n) = 1 and T(n+1,n) = (n+1) for n>=0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 16, 10, 3, 1, 127, 78, 21, 4, 1, 1363, 832, 216, 36, 5, 1, 18628, 11342, 2901, 460, 55, 6, 1, 311250, 189286, 48081, 7456, 840, 78, 7, 1, 6173791, 3752320, 949800, 145660, 15955, 1386, 105, 8, 1, 142190703, 86392756, 21826470, 3327340

Offset: 0

Paul D. Hanna, May 01 2005

Column 0 is A082161 (offset 1). Column 1 is (1/2)*A102087. Row sums form A106209.

			Triangle T begins:
1;
1,1;
3,2,1;
16,10,3,1;
127,78,21,4,1;
1363,832,216,36,5,1;
18628,11342,2901,460,55,6,1;
311250,189286,48081,7456,840,78,7,1;
6173791,3752320,949800,145660,15955,1386,105,8,1; ...
Matrix inverse T^-1 begins:
1;
-1,1;
-1,-2,1;
-3,-4,-3,1;
-16,-20,-9,-4,1;
-127,-156,-63,-16,-5,1;
-1363,-1664,-648,-144,-25,-6,1;
-18628,-22684,-8703,-1840,-275,-36,-7,1; ...
where [T^-1](n,k) = -(k+1)*T(n-1,k) when (n-1)>=k>=0.
G.f. for column 0: 1 = 1(1-x) + 1*x*(1-x)(1-2x) +
3*x^2*(1-x)(1-2x)(1-3x) + ... +
T(n,0)*x^n*(1-x)(1-2x)(1-3x)*..*(1-(n+1)*x) + ...
G.f. for column 1: 1 = 1(1-2x) + 2*x*(1-2x)(1-3x) +
10*x^2*(1-2x)(1-3x)(1-4x) + ... +
T(n+1,1)*x^n*(1-2x)(1-3x)(1-4x)*..*(1-(n+2)*x) + ...
G.f. for column 2: 1 = 1(1-3x) + 3*x*(1-3x)(1-4x) +
21*x^2*(1-3x)(1-4x)(1-5x) + ... +
T(n+2,2)*x^n*(1-3x)(1-4x)(1-5x)*..*(1-(n+3)*x) + ...

Cf. A102086, A082161, A106209.

PARI
```
T(n,k)=if(n
				
```

T(n,k)=local(A=matrix(1,1),B);A[1,1]=1; for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=j,if(j==1,B[i,j]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]));));A=B);return(A[n+1,k+1]/(k+1))

T(n, k) = A102086(n, k)/(k+1) for n>=0, k>=0. T(n, 0) = A082161(n) for n>0, with T(0, 0) = 1. G.f. for column k: 1 = Sum_{n>=0} T(n+k, k)*x^n*prod_{j=1, n+1} (1-(j+k)*x).

A106210 Triangular matrix T, read by rows, that satisfies: [T^-1](n,k) = -k^2T(n-2,k) when (n-2)>=k>=0, with T(n,n) = 1 and T(n+1,n) = (2n+1) for n>=0.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 16, 16, 5, 1, 127, 127, 39, 7, 1, 1363, 1363, 416, 72, 9, 1, 18628, 18628, 5671, 967, 115, 11, 1, 311250, 311250, 94643, 16027, 1864, 168, 13, 1, 6173791, 6173791, 1876160, 316600, 36415, 3191, 231, 15, 1, 142190703, 142190703

Offset: 0

Paul D. Hanna, May 01 2005

Both column 0 and column 1 form A082161. Row sums form A106211.

			Triangle T begins:
1;
1,1;
3,3,1;
16,16,5,1;
127,127,39,7,1;
1363,1363,416,72,9,1;
18628,18628,5671,967,115,11,1;
311250,311250,94643,16027,1864,168,13,1;
6173791,6173791,1876160,316600,36415,3191,231,15,1; ...
Matrix inverse T^-1 begins:
1;
-1,1;
0,-3,1;
0,-1,-5,1;
0,-3,-4,-7,1;
0,-16,-20,-9,-9,1;
0,-127,-156,-63,-16,-11,1;
0,-1363,-1664,-648,-144,-25,-13,1;
0,-18628,-22684,-8703,-1840,-275,-36,-15,1; ...
where [T^-1](n,k) = -k^2*T(n-2,k) when (n-2)>=k>=0.
G.f. for column 0: 1/(1-0x) = 1*(1-1x) + 1*x*(1-1x)(1-2x) +
3*x^2*(1-1x)(1-2x)(1-3x) + 16*x^3*(1-1x)(1-2x)(1-3x)(1-4x) + ...
+ T(n,0)*x^n*(1-1x)(1-2x)*..*(1-(n+1)*x) + ...
G.f. for column 1: 1/(1-1x) = 1*(1-2x) + 3*x*(1-2x)(1-3x) +
16*x^2*(1-2x)(1-3x)(1-4x) + 127*x^3*(1-2x)(1-3x)(1-4x)(1-5x) + ...
+ T(n+1,1)*x^n*(1-2x)(1-3x)*..*(1-(n+2)*x) + ...
G.f. for column 2: 1/(1-2x) = 1*(1-3x) + 5*x*(1-3x)(1-4x) +
39*x^2*(1-3x)(1-4x)(1-5x) + 416*x^3*(1-3x)(1-4x)(1-5x)(1-6x) + ...
+ T(n+2,2)*x^n*(1-3x)(1-4x)*..*(1-(n+3)*x) + ...

Cf. A102086, A082161, A106211.

PARI
```
T(n,k)=if(n
				
```

T(n,k)=local(A=matrix(1,1),B);A[1,1]=1; for(m=2,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=j,if(j==1,B[i,j]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]));));A=B); return(if(k==0,if(n==0,1,A[n+1,k+1]),A[n+1,k]/k^2))

T(n, k) = A102086(n, k)/(k+1) for n>=0, k>=0. T(n, 0) = A082161(n) for n>0, with T(0, 0) = 1. G.f. for column k: 1/(1-k*x) = Sum_{n>=0} T(n+k, k)*x^n*prod_{j=1, n+1} (1-(j+k)*x).

cp's OEIS Frontend

A102087 Column 1 of triangular matrix A102086, which shifts upward to exclude the main diagonal under matrix square.

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A102088 Row sums of triangular matrix A102086, which shifts upward to exclude the main diagonal under matrix square.

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A082161 Number of deterministic completely defined initially connected acyclic automata with 2 inputs and n transient unlabeled states (and a unique absorbing state).

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A102098 Triangular matrix, read by rows, that satisfies: T(n,k) = [T^3](n-1,k) when n>k>=0, with T(n,n) = (n+1).

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A102316 Triangle, read by rows, where T(n,k) = T(n,k-1) + (k+1)*T(n-1,k) for n>k>0, T(n,0)=1 and T(n,n) = T(n,n-1) for n>=0.

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A102916 Triangle, read by rows, where the antidiagonals are formed by interleaving the rows of triangle A102098 with the rows of its matrix square (A102920).

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A102101 Triangular matrix, read by rows, that satisfies: T(n,k) = [T^4](n-1,k) when n>k>=0, with T(n,n) = (n+1).

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A102320 Triangular matrix, read by rows, that satisfies: T(n,k) = [T^2](n-1,k) when n>k>=0, with T(n,n) = (2*n+1).

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A106210 Triangular matrix T, read by rows, that satisfies: [T^-1](n,k) = -k^2T(n-2,k) when (n-2)>=k>=0, with T(n,n) = 1 and T(n+1,n) = (2n+1) for n>=0.