A113084 -id:A113084 - cp's OEIS frontend

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).

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])}

A113087 Row sums 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, 8, 68, 1252, 51724, 5000468, 1176844012, 696653833108, 1063944196415276, 4273738134967763028, 45828521528368595280876, 1327368163105850592309345044, 104832927462400739554960944906668

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Cf. A113084.

{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(sum(k=1,n+1,M[n+1,k]))}

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.

Original entry on oeis.org

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]))}

A113086 Column 1 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, 4, 33, 586, 23299, 2166800, 490872957, 280082001078, 412989850899863, 1604627387895595612, 16673986511766782602033, 468802076619022792385223170, 35999655498002937223238590342603

Offset: 1

Paul D. Hanna, Oct 14 2005

nonn

Cf. A113084.

{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(if(n<1,0,M[n+1,2]))}

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])}

A113106 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^5](n-1,k-1) + [T^5](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^5 is the matrix 5th power of T.

Original entry on oeis.org

1, 1, 1, 5, 6, 1, 85, 115, 31, 1, 4985, 7420, 2590, 156, 1, 1082905, 1744965, 723370, 62090, 781, 1, 930005021, 1601759426, 752616215, 82390620, 1532715, 3906, 1, 3306859233805, 6024941167511, 3117415999361, 409321203715, 10025307495

Offset: 0

Paul D. Hanna, Oct 14 2005

Column 0 of the matrix power p, T^p, equals the number of 5-tournament sequences having initial term p (see A113103 for definitions).

			Triangle begins:
1;
1,1;
5,6,1;
85,115,31,1;
4985,7420,2590,156,1;
1082905,1744965,723370,62090,781,1;
930005021,1601759426,752616215,82390620,1532715,3906,1;
Matrix 4th power T^4 (A113112) begins:
1;
4,1;
56,24,1;
2704,1576,124,1;
481376,346624,39376,624,1; ...
where column 0 equals A113113.
Matrix 5th power T^5 (A113114) begins:
1;
5,1;
85,30,1;
4985,2435,155,1;
1082905,662060,61310,780,1;
930005021,671754405,80861810,1528810,3905,1; ...
where adjacent sums in row n of T^5 forms row n+1 of T.

Cf. A097710, A113084, A113095; A113103, A113107 (column 0), A113108 (T^2), A113110 (T^3), A113112 (T^4), A113112 (T^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^5)[r-1,c-1])+(M^5)[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^5] and for all integer p>=1: GF[T^p] = 1 + x*Sum_{j=1..p} GF[T^(p+4*j)] + x*y*GF[T^(5*p)].

A113095 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^4](n-1,k-1) + [T^4](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^4 is the matrix 4th power of T.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 46, 66, 21, 1, 1504, 2398, 978, 85, 1, 146821, 255113, 122914, 14962, 341, 1, 45236404, 84425001, 46001193, 7046354, 235122, 1365, 1, 46002427696, 91159696960, 54661544301, 9933169553, 432627794, 3738738, 5461, 1

Offset: 0

Paul D. Hanna, Oct 14 2005

Column 0 of the matrix power p, T^p, equals the number of 4-tournament sequences having initial term p (see A113092 for definitions).

			Triangle T begins:
  1;
  1,1;
  4,5,1;
  46,66,21,1;
  1504,2398,978,85,1;
  146821,255113,122914,14962,341,1;
  45236404,84425001,46001193,7046354,235122,1365,1; ...
Matrix third power T^3 (A113099) begins:
  1;
  3,1;
  27,15,1;
  693,513,63,1;
  52812,47619,8289,255,1; ...
 where column 0 equals A113100.
Matrix 4th power T^4 (A113101) begins:
  1;
  4,1;
  46,20,1;
  1504,894,84,1;
  146821,108292,14622,340,1;
  45236404,39188597,6812596,233758,1364,1; ...
 where adjacent sums in row n of T^4 forms row n+1 of T.

Cf. A097710, A113084, A113106; A113092, A113096 (column 0), A113097 (T^2), A113099 (T^3), A113101 (T^4).

{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^4)[r-1,c-1])+(M^4)[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^4] and for all integer p>=1: GF[T^p] = 1 + x*Sum_{j=1..p} GF[T^(p+3*j)] + x*y*GF[T^(4*p)].

A113081 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 3-tournament sequences, for n>=1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 21, 10, 3, 1, 0, 331, 114, 21, 4, 1, 0, 11973, 2970, 331, 36, 5, 1, 0, 1030091, 182402, 11973, 724, 55, 6, 1, 0, 218626341, 27392682, 1030091, 33476, 1345, 78, 7, 1, 0, 118038692523, 10390564242, 218626341, 3697844, 75695, 2246

Offset: 0

Paul D. Hanna, Oct 14 2005

A 3-tournament sequence is an increasing sequence of positive integers (t_1,t_2,...) such that t_1 = p, t_i = p (mod 2) and t_{i+1} <= 3*t_i, where p>=1. This is the table of 3-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,...
0,3,10,21,36,55,78,105,136,171,210,...
0,21,114,331,724,1345,2246,3479,5096,7149,...
0,331,2970,11973,33476,75695,148926,265545,440008,...
0,11973,182402,1030091,3697844,10204145,23694838,...
0,1030091,27392682,218626341,1011973796,3416461455,...
0,218626341,10390564242,118038692523,706848765844,...
0,118038692523,10210795262650,166013096151621,...

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

Cf. A113084, A113085 (column 1), A113089 (column 2); tables: A093729 (2-tournaments), A113092 (4-tournaments), A113103 (5-tournaments).

/* Generalized Cook-Kleber Recurrence */ T(n,k,q=3)=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))))))

/* Matrix Power Recurrence (Paul D. Hanna) */ T(n,k,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^k)[n+1,1])

For n>=k>0: T(n, k) = Sum_{j=1..k} T(n-1, k+2*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 A113084, which satisfies the matrix recurrence: A113084(n, k) = [A113084^3](n-1, k-1) + [A113084^3](n-1, k) for n>k>=0.

cp's OEIS Frontend

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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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).

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A113087 Row sums 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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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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A113086 Column 1 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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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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A113106 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^5](n-1,k-1) + [T^5](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^5 is the matrix 5th power of T.

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A113095 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^4](n-1,k-1) + [T^4](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^4 is the matrix 4th power of T.

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A113081 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 3-tournament sequences, for n>=1.

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