A113081 -id:A113081 - cp's OEIS frontend

A113082 Main diagonal of square table A113081; also, a(n) equals the n-th term in column 0 of the matrix n-th power of triangle A113084.

1, 1, 10, 331, 33476, 10204145, 9378590446, 26026690264407, 218132378185337416, 5518274388618175447069, 421034872020570533423509010, 96809747319527667989371938562883

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Cf. A113081, A113084.

{a(n,q=3)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^q)[r-1,c-1])+(M^q)[r-1,c]))); return((M^n)[n+1,1])}

A113083 Diagonal of square table A113081; also, a(n) equals the n-th term in column 0 of the matrix (n+1)-th power of triangle A113084.

Original entry on oeis.org

1, 2, 21, 724, 75695, 23694838, 22239639177, 62747494950248, 532868670719193651, 13624738004791751175370, 1048678107774203901392276461, 242892250870416811233766661498812

Offset: 1

Paul D. Hanna, Oct 14 2005

nonn

Cf. A113081, A113084.

{a(n,q=3)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^q)[r-1,c-1])+(M^q)[r-1,c]))); return(if(n<1,0,(M^n)[n,1]))}

A113089 Number of 3-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 2 and t_i = 2 (mod 2) and t_{i+1} <= 3*t_i for 1

Original entry on oeis.org

1, 2, 10, 114, 2970, 182402, 27392682, 10390564242, 10210795262650, 26494519967902114, 184142934938620227530, 3466516611360924222460082, 178346559667060145108789818842, 25264074391478558474014952210052802

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Column 0 of triangle A113088; A113088 is the matrix square of triangle A113084, which satisfies the matrix recurrence: A113084(n,k) = [A113084^3](n-1,k-1) + [A113084^3](n-1,k). Also equals column 2 of square table A113081.

			The tree of 3-tournament sequences of even integer
descendents of a node labeled (2) begins:
[2]; generation 1: 2->[4,6];
generation 2: 4->[6,8,10,12], 6->[8,10,12,14,16,18]; ...
Then a(n) gives the number of nodes in generation n.
Also, a(n+1) = sum of labels of nodes in generation n.

T. D. Noe, Table of n, a(n) for n=0..30
M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

Cf. A008934, A113077, A113078, A113079, A113085, A113096, A113098, A113100, A113107, A113109, A113111, A113113.

{a(n)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^3)[r-1,c-1])+(M^3)[r-1,c]))); return((M^2)[n+1,1])}

A113084 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^3](n-1,k-1) + [T^3](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^3 is the matrix third power of T.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 21, 33, 13, 1, 331, 586, 294, 40, 1, 11973, 23299, 13768, 2562, 121, 1, 1030091, 2166800, 1447573, 333070, 22569, 364, 1, 218626341, 490872957, 361327779, 97348117, 8466793, 200931, 1093, 1, 118038692523, 280082001078

Offset: 0

Paul D. Hanna, Oct 14 2005

Column 0 of the matrix power p, T^p, equals the number of 3-tournament sequences having initial term p.

			Triangle T begins:
1;
1,1;
3,4,1;
21,33,13,1;
331,586,294,40,1;
11973,23299,13768,2562,121,1;
1030091,2166800,1447573,333070,22569,364,1; ...
Matrix square T^2 (A113088) begins:
1;
2,1;
10,8,1;
114,118,26,1;
2970,3668,1108,80,1;
182402,257122,96416,9964,242,1; ...
where column 0 equals A113089.
Matrix cube T^3 (A113090) begins:
1;
3,1;
21,12,1;
331,255,39,1;
11973,11326,2442,120,1;
1030091,1136709,310864,22206,363,1; ...
where adjacent sums in row n of T^3 forms row n+1 of T.

Cf. A113081; A097710, A113095, A113106; A113085 (column 0), A113088 (T^2), A113087 (row sums).

{T(n,k)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^3)[r-1,c-1])+(M^3)[r-1,c]))); return(M[n+1,k+1])}

Let GF[T] denote the g.f. of triangular matrix T. Then GF[T] = 1 + x*(1+y)*GF[T^3] and for all integer p>=1: GF[T^p] = 1 + x*Sum_{j=1..p} GF[T^(p+2*j)] + x*y*GF[T^(3*p)].

A093729 Square table T, read by antidiagonals, where T(n,k) gives the number of n-th generation descendents of a node labeled (k) in the tree of tournament sequences.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 7, 7, 3, 1, 0, 41, 41, 15, 4, 1, 0, 397, 397, 123, 26, 5, 1, 0, 6377, 6377, 1656, 274, 40, 6, 1, 0, 171886, 171886, 36987, 4721, 515, 57, 7, 1, 0, 7892642, 7892642, 1391106, 134899, 10810, 867, 77, 8, 1, 0, 627340987, 627340987, 89574978, 6501536, 376175, 21456, 1351, 100, 9, 1

Offset: 0

Paul D. Hanna, Apr 14 2004; revised Oct 14 2005

Column 1, of array T and antidiagonals, equals A008934, which is the number of tournament sequences.

A tournament sequence is an increasing sequence of positive integers (t_1,t_2,...) such that t_1 = 1 and t_{i+1} <= 2*t_i, where integer k>1.

			Array begins:
  1,      1,       1,       1,       1,      1,      1,     1,     1, ...],
  0,      1,       2,       3,       4,      5,      6,     7,     8, ...],
  0,      2,       7,      15,      26,     40,     57,    77,   100, ...],
  0,      7,      41,     123,     274,    515,    867,  1351,  1988, ...],
  0,     41,     397,    1656,    4721,  10810,  21456, 38507, 64126, ...],
  0,    397,    6377,   36987,  134899, 376175, 880032, .................],
  0,   6377,  171886, 1391106, 6501536, ...],
  0, 171886, 7892642, .....................];
Antidiagonals begin as:
  1;
  0,      1;
  0,      1,      1;
  0,      2,      2,     1;
  0,      7,      7,     3,    1;
  0,     41,     41,    15,    4,   1;
  0,    397,    397,   123,   26,   5,   1;
  0,   6377,   6377,  1656,  274,  40,   6,   1;
  0, 171886, 171886, 36987, 4721, 515,  57,   7,   1;

G. C. Greubel, Antidiagonals n = 0..50, flattened
M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.
Michael Somos, A functional power series equation, Mathematics StackExchange answer.

Cf. A008934, A097710, A113080, A113081, A113092, A113103.

Cf. A008934 (column k=1 of array and antidiagonals), A093730 (antidiagonal row sums).

t[n_?Negative, ] = 0; t[0, ] = 1; t[n_, k_] /; k <= n := t[n, k] = t[n, k - 1] - t[n-1, k] + t[n - 1, 2 k - 1] + t[n - 1, 2 k]; t[n_, k_] := t[n, k] = Sum[(-1)^(j - 1)*Binomial[n + 1, j]*t[n, k - j], {j, 1, n + 1}]; Flatten[Table[t[i - k, k - 1], {i, 10}, {k, i}]] (* Jean-François Alcover, May 31 2011, after PARI prog. *)

{T(n,k)=if(n<0,0,if(n==0,1,if(k==0,0, if(k<=n,T(n,k-1)-T(n-1,k)+T(n-1,2*k-1)+T(n-1,2*k), sum(j=1,n+1, (-1)^(j-1)*binomial(n+1,j)*T(n,k-j))))))}

{a(n, m) = my(A=1); for(k=1, n, A = (A - q^k * r * subst( subst(A, q, q^2), r, r^2)) / (1-q)); subst(subst(A, r, q^(m-1)), q, 1)}; /* Michael Somos, Jun 19 2017 */

@CachedFunction
def T(n, k):
    if n<0: return 0
    elif n==0: return 1
    elif k==0: return 0
    elif kA093729(n,k): return T(n-k,k)
flatten([[A093729(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Feb 22 2024

T(0, k)=1 for k>=0, T(n, 0)=0 for n>=1; else T(n, k) = T(n, k-1) - T(n-1, k) + T(n-1, 2*k-1) + T(n-1, 2*k) for k<=n; else T(n, k) = Sum_{j=1..n+1} (-1)^(j-1)*C(n+1, j)*T(n, k-j) for k>n (Cook-Kleber).
Column k of T equals column 0 of the matrix k-th power of triangle A097710, which satisfies the matrix recurrence: A097710(n, k) = [A097710^2](n-1, k-1) + [A097710^2](n-1, k) for n>k>=0.
Sum_{k=0..n} T(n-k, k) = A093730(n) (antidiagonal row sums).

A113092 Square table T, read by antidiagonals, where T(n,k) gives the number of n-th generation descendents of a node labeled (k) in the tree of 4-tournament sequences.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 46, 13, 3, 1, 0, 1504, 242, 27, 4, 1, 0, 146821, 13228, 693, 46, 5, 1, 0, 45236404, 2241527, 52812, 1504, 70, 6, 1, 0, 46002427696, 1237069018, 12628008, 146821, 2780, 99, 7, 1, 0, 159443238441379, 2305369985312, 9924266772

Offset: 0

Paul D. Hanna, Oct 14 2005

A 4-tournament sequence is an increasing sequence of positive integers (t_1,t_2,...) such that t_1 = p, t_i = p (mod 3) and t_{i+1} <= 4*t_i, where p>=1. This is the table of 4-tournament sequences when the starting node has label p = k for column k>=1.

			Table begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,...
0,1,2,3,4,5,6,7,8,9,10,11,...
0,4,13,27,46,70,99,133,172,216,265,...
0,46,242,693,1504,2780,4626,7147,10448,14634,...
0,1504,13228,52812,146821,330745,648999,1154923,1910782,...
0,146821,2241527,12628008,45236404,124626530,289031301,...
0,45236404,1237069018,9924266772,46002427696,155367674020,...
0,46002427696,2305369985312,26507035453923,159443238441379,...
0,159443238441379,14874520949557933,246323730279500082,...

M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

Cf. A113095, A113096 (column 1), A113098 (column 2), A113100 (column 2); Tables: A093729 (2-tournaments), A113081 (3-tournaments), A113103 (5-tournaments); diagonals: A113093, A113094.

/* Generalized Cook-Kleber Recurrence */
{T(n,k,q=4)=if(n==0,1,if(n<0||k<=0,0,if(n==1,k, if(n>=k,sum(j=1,k,T(n-1,k+(q-1)*j)), sum(j=1,n+1,(-1)^(j-1)*binomial(n+1,j)*T(n,k-j))))))}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))

/* Matrix Power Recurrence (Paul D. Hanna) */
{T(n,k,q=4)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^q)[r-1,c-1])+(M^q)[r-1,c]))); return((M^k)[n+1,1])}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))

For n>=k>0: T(n, k) = Sum_{j=1..k} T(n-1, k+3*j); else for k>n>0: T(n, k) = Sum_{j=1..n+1}(-1)^(j-1)*C(n+1, j)*T(n, k-j); with T(0, k)=1 for k>=0. Also, column k of T equals column 0 of the matrix k-th power of triangle A113095, which satisfies the matrix recurrence: A113095(n, k) = [A113095^4](n-1, k-1) + [A113095^4](n-1, k) for n>k>=0.

A113103 Square table T, read by antidiagonals, where T(n,k) gives the number of n-th generation descendents of a node labeled (k) in the tree of 5-tournament sequences.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 2, 1, 0, 85, 16, 3, 1, 0, 4985, 440, 33, 4, 1, 0, 1082905, 43600, 1251, 56, 5, 1, 0, 930005021, 16698560, 173505, 2704, 85, 6, 1, 0, 3306859233805, 26098464448, 94216515, 481376, 4985, 120, 7, 1, 0, 50220281721033905

Offset: 0

Paul D. Hanna, Oct 14 2005

A 5-tournament sequence is an increasing sequence of positive integers (t_1,t_2,...) such that t_1 = p, t_i = p (mod 4) and t_{i+1} <= 5*t_i, where p>=1. This is the table of 5-tournament sequences when the starting node has label p = k for column k>=1.

			Table begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,...
0,1,2,3,4,5,6,7,8,9,10,11,...
0,5,16,33,56,85,120,161,208,261,320,...
0,85,440,1251,2704,4985,8280,12775,18656,26109,...
0,4985,43600,173505,481376,1082905,2122800,3774785,6241600,...
0,1082905,16698560,94216515,337587520,930005021,2156566656,...
0,930005021,26098464448,210576669921,978162377600,...
0,3306859233805,172513149018752,2002383115518243,...
0,50220281721033905,4938593053649344000,82856383278525698433,...

M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

Cf. A113106, A113107 (column 1), A113109 (column 2), A113111 (column 3), A113113 (column 4); Tables: A093729 (2-tournaments), A113081 (3-tournaments), A113092 (4-tournaments).

/* Generalized Cook-Kleber Recurrence */
{T(n,k,q=5)=if(n==0,1,if(n<0||k<=0,0,if(n==1,k, if(n>=k,sum(j=1,k,T(n-1,k+(q-1)*j)), sum(j=1,n+1,(-1)^(j-1)*binomial(n+1,j)*T(n,k-j))))))}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))

/* Matrix Power Recurrence (Paul D. Hanna) */
{T(n,k,q=5)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^q)[r-1,c-1])+(M^q)[r-1,c]))); (M^k)[n+1,1]}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))

For n>=k>0: T(n, k) = Sum_{j=1..k} T(n-1, k+4*j); else for k>n>0: T(n, k) = Sum_{j=1..n+1}(-1)^(j-1)*C(n+1, j)*T(n, k-j); with T(0, k)=1 for k>=0. Column k of T equals column 0 of the matrix k-th power of triangle A113106, which satisfies the matrix recurrence: A113106(n, k) = [A113106^5](n-1, k-1) + [A113106^5](n-1, k) for n>k>=0.

A113088 Triangle T, read by rows, equal to the matrix square of triangle A113084, which satisfies the recurrence: A113084(n,k) = [A113084^3](n-1,k-1) + [A113084^3](n-1,k).

Original entry on oeis.org

1, 2, 1, 10, 8, 1, 114, 118, 26, 1, 2970, 3668, 1108, 80, 1, 182402, 257122, 96416, 9964, 242, 1, 27392682, 42821472, 18871894, 2501468, 89182, 728, 1, 10390564242, 17650889358, 8826033518, 1412198686, 65914154, 799714, 2186, 1

Offset: 0

Paul D. Hanna, Oct 14 2005

			Triangle begins:
1;
2,1;
10,8,1;
114,118,26,1;
2970,3668,1108,80,1;
182402,257122,96416,9964,242,1;
27392682,42821472,18871894,2501468,89182,728,1; ...

Cf. A113084, A113081.

{T(n,k)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^3)[r-1,c-1])+(M^3)[r-1,c]))); return((M^2)[n+1,k+1])}

A113090 Triangle T, read by rows, equal to the matrix cube of triangle A113084, which satisfies the recurrence: A113084(n,k) = [A113084^3](n-1,k-1) + [A113084^3](n-1,k).

Original entry on oeis.org

1, 3, 1, 21, 12, 1, 331, 255, 39, 1, 11973, 11326, 2442, 120, 1, 1030091, 1136709, 310864, 22206, 363, 1, 218626341, 272246616, 89081163, 8266954, 199839, 1092, 1, 118038692523, 162043308555, 61099562421, 6923071251, 220482175, 1796349

Offset: 0

Paul D. Hanna, Oct 14 2005

			Triangle begins:
1;
3,1;
21,12,1;
331,255,39,1;
11973,11326,2442,120,1;
1030091,1136709,310864,22206,363,1;
218626341,272246616,89081163,8266954,199839,1092,1; ...

Cf. A113084, A113081, A113091 (column 1).

{T(n,k)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^3)[r-1,c-1])+(M^3)[r-1,c]))); return((M^3)[n+1,k+1])}

A113080 Square table, read by antidiagonals, where T(n,k) equals the number of k-tournament sequences of length n for k>=1, with T(0,k) = 1 for k>=1 and T(n,1) = 0 for n>0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 7, 10, 3, 1, 0, 41, 114, 27, 4, 1, 0, 397, 2970, 693, 56, 5, 1, 0, 6377, 182402, 52812, 2704, 100, 6, 1, 0, 171886, 27392682, 12628008, 481376, 8125, 162, 7, 1, 0, 7892642, 10390564242, 9924266772, 337587520, 2918750, 20502, 245, 8

Offset: 1

Paul D. Hanna, Oct 14 2005

A k-tournament sequence is an increasing sequence of positive integers (t_1,t_2,...) such that t_1 = p, t_i = p (mod k-1) and t_{i+1} <= k*t_i, where k>1, p>=1. This is the table of k-tournament sequences when the starting node has label p = 1 for k>=1.

			Table begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,...
0,1,2,3,4,5,6,7,8,9,10,11,...
0,2,10,27,56,100,162,245,352,486,650,...
0,7,114,693,2704,8125,20502,45619,92288,173259,...
0,41,2970,52812,481376,2918750,13399506,50216915,...
0,397,182402,12628008,337587520,4976321250,48633051942,...
0,6377,27392682,9924266772,978162377600,42197834315625,...
0,171886,10390564242,26507035453923,12088945462984960,...
0,7892642,10210795262650,246323730279500082,...

M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

Columns: A008934 (k=2), A113089 (k=3), A113100 (k=4), A113113 (k=5); related tables: A093729 (k=2), A113081 (k=3), A113092 (k=4), A113103 (k=5).

{T(n,k)=local(M=matrix(n+1,n+1));for(r=1,n+1, for(c=1,r, M[r,c]=if(r==c,1,if(c>1,(M^k)[r-1,c-1])+(M^k)[r-1,c]))); return((M^(k-1))[n+1,1])}

cp's OEIS Frontend

A113082 Main diagonal of square table A113081; also, a(n) equals the n-th term in column 0 of the matrix n-th power of triangle A113084.

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A113083 Diagonal of square table A113081; also, a(n) equals the n-th term in column 0 of the matrix (n+1)-th power of triangle A113084.

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A113089 Number of 3-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 2 and t_i = 2 (mod 2) and t_{i+1} <= 3*t_i for 1

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A113084 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^3](n-1,k-1) + [T^3](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^3 is the matrix third power of T.

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A093729 Square table T, read by antidiagonals, where T(n,k) gives the number of n-th generation descendents of a node labeled (k) in the tree of tournament sequences.

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A113092 Square table T, read by antidiagonals, where T(n,k) gives the number of n-th generation descendents of a node labeled (k) in the tree of 4-tournament sequences.

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A113103 Square table T, read by antidiagonals, where T(n,k) gives the number of n-th generation descendents of a node labeled (k) in the tree of 5-tournament sequences.

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A113088 Triangle T, read by rows, equal to the matrix square of triangle A113084, which satisfies the recurrence: A113084(n,k) = [A113084^3](n-1,k-1) + [A113084^3](n-1,k).

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