A102098 -id:A102098 - cp's OEIS frontend

A102920 Triangular matrix, read by rows, equal to the matrix square of A102098.

1, 3, 4, 36, 40, 9, 1036, 1128, 189, 16, 56355, 61120, 9720, 576, 25, 5045370, 5466320, 857466, 47040, 1375, 36, 679409158, 735847800, 114915375, 6155008, 163500, 2808, 49, 129195427716, 139910204080, 21813099606, 1158059520, 29767000, 458136

Offset: 0

Paul D. Hanna, Jan 21 2005

Column 0 is A102921. Column 1 is A102922.

			Rows begin:
[1],
[3,4],
[36,40,9],
[1036,1128,189,16],
[56355,61120,9720,576,25],
[5045370,5466320,857466,47040,1375,36],
[679409158,735847800,114915375,6155008,163500,2808,49],...

Cf. A102098, A102921, A102922, A102916.

{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,B[i,j]=(A^3)[i-1,j]);));A=B); return((A^2)[n+1,k+1])}

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).

A102099 Column 1 of triangular matrix A102098, which shifts upward to exclude the main diagonal under matrix cube.

Original entry on oeis.org

0, 2, 8, 152, 6200, 442552, 49399320, 8003532512, 1784040237288, 525504809786112, 198213959637435608, 93352856625931514024, 53776417402985961020144, 37244016639064540041311632

Offset: 0

Paul D. Hanna, Dec 29 2004

nonn

Column 0 of A102098 is A082162.

Cf. A102098, A082162, A102100.

{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^3)[i-1,1], B[i,j]=(A^3)[i-1,j]));));A=B); return(A[n+1,2])}

A102921 Column 0 of triangle A102920, which equals the matrix square of A102098.

Original entry on oeis.org

1, 3, 36, 1036, 56355, 5045370, 679409158, 129195427716, 33114233390505, 11038606786054201, 4652371578279864792, 2423023045813285312020, 1530233703568825263174101, 1153422053136775523883308988

Offset: 0

Paul D. Hanna, Jan 21 2005

nonn

Equals the bisection of A102917. Triangle A102098 shifts each column up 1 row under matrix cube.

Cf. A102920, A102098, A102917.

{a(n)=if(n==0,2,polcoeff(2-sum(k=0,n-1,a(k)*x^k*prod(j=2,k\2+2,1-j*x +x*O(x^n))),n))} /* at even-indices n=0,2,4,.. */

A102100 Row sums of triangular matrix A102098, which shifts upward to exclude the main diagonal under matrix cube.

Original entry on oeis.org

1, 3, 18, 322, 12979, 924628, 103158338, 16710522378, 3724631345923, 1097090407192683, 413803244841678483, 194887616017161359389, 112265654949194591311618, 77751843768367000711311005

Offset: 0

Paul D. Hanna, Dec 29 2004

nonn

Cf. A102098, A102099.

{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^3)[i-1,1], B[i,j]=(A^3)[i-1,j]));));A=B); return(sum(k=0,n,A[n+1,k+1]))}

A102922 Column 1 of triangle A102920, which equals the matrix square of A102098.

Original entry on oeis.org

0, 4, 40, 1128, 61120, 5466320, 735847800, 139910204080, 35858685086352, 11953187179149408, 5037776918810353960, 2623732639426967662648, 1656984556235159516822400, 1248959074762601252295551168

Offset: 0

Paul D. Hanna, Jan 21 2005

nonn

Equals the odd bisection of A102917. Triangle A102098 shifts each column up 1 row under matrix cube.

Cf. A102920, A102098, A102917.

{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^3)[i-1,1], B[i,j]=(A^3)[i-1,j]));));A=B);return((A^2)[n+1,2])}

A102923 Row sums of triangle A102920, which equals the matrix square of A102098.

Original entry on oeis.org

1, 7, 85, 2369, 127796, 11417607, 1536493698, 292107021267, 74862835208823, 24954353268384100, 10517125257007205287, 5477412008465124456814, 3459179319447147792978276, 2607366906177104506271124036

Offset: 0

Paul D. Hanna, Jan 21 2005

nonn

Triangle A102098 shifts each column up 1 row under matrix cube.

Cf. A102920, A102098.

{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,B[i,j]=(A^3)[i-1,j]);));A=B); return(sum(k=0,n,(A^2)[n+1,k+1]))}

A082162 Number of deterministic completely defined initially connected acyclic automata with 3 inputs and n transient unlabeled states (and a unique absorbing state).

Original entry on oeis.org

1, 7, 139, 5711, 408354, 45605881, 7390305396, 1647470410551, 485292763088275, 183049273155939442, 86211400693272461866

Offset: 1

Valery A. Liskovets, Apr 09 2003

Coefficients T_3(n,k) form the array A082170. These automata have no nontrivial automorphisms (by states).

R. Bacher, C. 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 (after Jean-François Alcover), Table of n, a(n) for n = 1..210
Manosij Ghosh Dastidar and Michael Wallner, Asymptotics of relaxed k-ary trees, arXiv:2404.08415 [math.CO], 2024. See p. 1.4.
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.

Cf. A082158, A082161.

Cf. A102098, A102400.

T[n_, k_] := T[n, k] = If[nJean-François Alcover, Dec 15 2014 *)

a(n) = c_3(n)/(n-1)! where c_3(n) = T_3(n, 1) - sum(binomial(n-1, j-1)*T_3(n-j, j+1)*c_3(j), j=1..n-1) and T_3(0, k) = 1, T_3(n, k) = sum(binomial(n, i)*(-1)^(n-i-1)*(i+k)^(3*n-3*i)*T_3(i, k), i=0..n-1), n>0.

Equals column 0 of triangle A102098. Also equals main diagonal of A102400: a(n) = A102098(n, 0) = A102400(n, n). - Paul D. Hanna, Jan 07 2005

More terms from Paul D. Hanna, Jan 07 2005

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.

A102917 Column 0 of triangle A102916.

Original entry on oeis.org

1, 1, 1, 3, 7, 36, 139, 1036, 5711, 56355, 408354, 5045370, 45605881, 679409158, 7390305396, 129195427716, 1647470410551, 33114233390505, 485292763088275, 11038606786054201, 183049273155939442, 4652371578279864792

Offset: 0

Paul D. Hanna, Jan 21 2005

nonn

Also equals the interleaving of A082162 with A102921, which equal column 0 of triangle A102098 and its matrix square (A102920), respectively.

			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)
+ 139*x^6*(1-x)(1-2x)(1-3x)(1-4x) + 1036*x^7*(1-x)(1-2x)(1-3x)(1-4x) + ...
+ 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) + ...

Cf. A102916, A082162, A102921, A102098, A102920.

{a(n)=if(n==0,1,polcoeff(1-sum(k=0,n-1,a(k)*x^k*prod(j=1,k\2+1,1-j*x+x*O(x^n))),n))}

G.f.: 1 = Sum_{n>=0}(a(2*n)+a(2*n+1)*x)*x^(2*n)*Product_{k=1..n+1}(1-k*x) where a(2*n)=A082162(n) and a(2*n+1)=A102921(n).

cp's OEIS Frontend

A102920 Triangular matrix, read by rows, equal to the matrix square of A102098.

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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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A102099 Column 1 of triangular matrix A102098, which shifts upward to exclude the main diagonal under matrix cube.

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A102921 Column 0 of triangle A102920, which equals the matrix square of A102098.

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A102100 Row sums of triangular matrix A102098, which shifts upward to exclude the main diagonal under matrix cube.

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A102922 Column 1 of triangle A102920, which equals the matrix square of A102098.

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A102923 Row sums of triangle A102920, which equals the matrix square of A102098.

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A082162 Number of deterministic completely defined initially connected acyclic automata with 3 inputs and n transient unlabeled states (and a unique absorbing state).

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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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