A113092 -id:A113092 - cp's OEIS frontend

A113093 Main diagonal of square table A113092; also, a(n) equals the n-th term in column 0 of the matrix n-th power of triangle A113095.

1, 1, 13, 693, 146821, 124626530, 426524622399, 5893207147435867, 328422072384464274577, 73719657441008064407836359, 66567306698774377126527799872190

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Cf. A113092, A113095.

{a(n,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^n)[n+1,1])}

A113094 Diagonal of square table A113092; also, a(n) equals the n-th term in column 0 of the matrix (n+1)-th power of triangle A113095.

Original entry on oeis.org

1, 2, 27, 1504, 330745, 289031301, 1011348629263, 14213347986246578, 802722082112213275116, 182118530044524172384716760, 165892108866362877173717099499469

Offset: 1

Paul D. Hanna, Oct 14 2005

nonn

Cf. A113092, A113095.

{a(n,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(if(n<1,0,(M^n)[n,1]))}

A113100 Number of 4-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 3 and t_i = 3 (mod 3) and t_{i+1} <= 4*t_i for 1

Original entry on oeis.org

1, 3, 27, 693, 52812, 12628008, 9924266772, 26507035453923, 246323730279500082, 8100479557816637139288, 954983717308947379891713642, 407790020849346203244152231395953

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Column 0 of triangle A113099; A113099 is the matrix cube of triangle A113095, which satisfies the matrix recurrence: A113095(n,k) = [A113095^4](n-1,k-1) + [A113095^4](n-1,k). Also equals column 3 of square table A113092.

			The tree of 4-tournament sequences of descendents of a node labeled (3) begins:
[3]; generation 1: 3->[6,9,12]; generation 2:
6->[9,12,15,18,21,24], 9->[12,15,18,21,24,27,30,33,36],
12->[15,18,21,24,27,30,33,36,39,42,45,48]; ...
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, A113089, A113096, A113098, 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^4)[r-1,c-1])+(M^4)[r-1,c]))); return((M^3)[n+1,1])}

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)].

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).

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.

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.

A113101 Triangle T, read by rows, equal to the matrix 4th power of triangle A113095, which satisfies the recurrence: A113095(n,k) = [A113095^4](n-1,k-1) + [A113095^4](n-1,k).

Original entry on oeis.org

1, 4, 1, 46, 20, 1, 1504, 894, 84, 1, 146821, 108292, 14622, 340, 1, 45236404, 39188597, 6812596, 233758, 1364, 1, 46002427696, 45157269264, 9504275037, 428894516, 3733278, 5460, 1, 159443238441379, 172969059719500

Offset: 0

Paul D. Hanna, Oct 14 2005

Column 0 equals A113096 shift left one place.

			Triangle begins:
1;
4,1;
46,20,1;
1504,894,84,1;
146821,108292,14622,340,1;
45236404,39188597,6812596,233758,1364,1;
46002427696,45157269264,9504275037,428894516,3733278,5460,1;

Cf. A113092 (table), A113096 (column 0).

{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^4)[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

A113093 Main diagonal of square table A113092; also, a(n) equals the n-th term in column 0 of the matrix n-th power of triangle A113095.

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A113094 Diagonal of square table A113092; also, a(n) equals the n-th term in column 0 of the matrix (n+1)-th power of triangle A113095.

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A113100 Number of 4-tournament sequences: a(n) gives the number of increasing sequences of n positive integers (t_1,t_2,...,t_n) such that t_1 = 3 and t_i = 3 (mod 3) and t_{i+1} <= 4*t_i for 1

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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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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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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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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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A113101 Triangle T, read by rows, equal to the matrix 4th power of triangle A113095, which satisfies the recurrence: A113095(n,k) = [A113095^4](n-1,k-1) + [A113095^4](n-1,k).

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