A093654 -id:A093654 - cp's OEIS frontend

A093655 First column of lower triangular matrix A093654.

1, 1, 1, 2, 1, 2, 2, 7, 1, 2, 2, 7, 2, 7, 7, 41, 1, 2, 2, 7, 2, 7, 7, 41, 2, 7, 7, 41, 7, 41, 41, 397, 1, 2, 2, 7, 2, 7, 7, 41, 2, 7, 7, 41, 7, 41, 41, 397, 2, 7, 7, 41, 7, 41, 41, 397, 7, 41, 41, 397, 41, 397, 397, 6377

Offset: 1

Paul D. Hanna, Apr 08 2004

nonn

Related to the number of tournament sequences (A008934).

a(n) equals the number of tournament sequences (A008934) of length A000120(n-1), which is the number of 1's in the binary expansion of n-1.

Cf. A008934, A093655.

Cf. A000120.

a(2^n) = A008934(n) for n>=0.

a(n) = A008934(A000120(n-1)) for n>=1.

A093657 2^(n-1)-th term of the row sums of triangle A093654.

Original entry on oeis.org

1, 2, 6, 28, 206, 2418, 45970, 1440746, 75840096, 6828414424, 1069361760254, 295609883371824, 146078092162147126, 130419475982163166640, 212257994312591826735888, 634463537260289571176650942

Offset: 1

Paul D. Hanna, Apr 08 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 1..86

Cf. A093654, A093656.
Related to the number of tournament sequences (A008934).
Cf. A097710, A008934.

T[n_, k_]:= T[n,k]= If[n<0 || k>n, 0, If[n==k, 1, If[k==0, Sum[T[n-1,j]*T[j,0], {j,0,n-1}], Sum[T[n-1,j]*(T[j,k-1]+T[j,k]), {j,0,n-1}] ]]]; (* T = A097710 *)
A093657[n_]:= A093657[n]= Sum[T[n,k], {k,0,n}];
Table[A093657[n], {n,0,30}] (* G. C. Greubel, Feb 21 2024 *)

@CachedFunction
def T(n, k): # T = A097710
    if n< 0 or k<0 or k>n: return 0
    elif k==n: return 1
    elif k==0: return sum(T(n-1,j)*T(j,0) for j in range(n))
    else: return sum(T(n-1, j)*(T(j, k-1)+T(j,k)) for j in range(n))
def A093657(n): return sum(T(n,k) for k in range(n+1))
[A093657(n) for n in range(31)] # G. C. Greubel, Feb 21 2024

a(n) = A093656(2^(n-1)) for n>=1.

a(n) = Sum_{k=0..n} A097710(n,k), row sums of triangle A097710.

A093656 Row sums of lower triangular matrix A093654.

Original entry on oeis.org

1, 2, 2, 6, 2, 6, 6, 28, 2, 6, 6, 28, 6, 28, 28, 206, 2, 6, 6, 28, 6, 28, 28, 206, 6, 28, 28, 206, 28, 206, 206, 2418, 2, 6, 6, 28, 6, 28, 28, 206, 6, 28, 28, 206, 28, 206, 206, 2418, 6, 28, 28, 206, 28, 206, 206, 2418, 28, 206, 206, 2418, 206, 2418, 2418, 45970

Offset: 1

Paul D. Hanna, Apr 08 2004

nonn

Cf. A093654.

a(2^n+1) = 2 for n>=0. a(2^n+2^m) = a(2^(m+1)) for n>m>=0.

A008934 Number of tournament sequences: sequences (a_1, a_2, ..., a_n) with a_1 = 1 such that a_i < a_{i+1} <= 2*a_i for all i.

Original entry on oeis.org

1, 1, 2, 7, 41, 397, 6377, 171886, 7892642, 627340987, 87635138366, 21808110976027, 9780286524758582, 7981750158298108606, 11950197013167283686587, 33046443615914736611839942, 169758733825407174485685959261, 1627880269212042994531083889564192

Offset: 0

Mauro Torelli (torelli(AT)hermes.mc.dsi.unimi.it), Jeffrey Shallit

Also number of Meeussen sequences of length n (see the Cook-Kleber reference).
Column 1 of triangle A093729. Also generated by the iteration procedure that constructs triangle A093654. - Paul D. Hanna, Apr 14 2004
a(n) is the number of sequences (u_1,u_2,...,u_n) of positive integers such that u_1=1 and u_i <= 1+ u_1+...+u_{i-1} for 2<=i<=n. For example, omitting parentheses and commas, a(3)=7 counts 111, 112, 113, 121, 122, 123, 124. The difference-between-successive-terms operator is a bijection from the title sequences to these sequences. For example, the tournament sequence (1, 2, 4, 5, 9, 16) bijects to (1,2,1,4,7). (To count tournament sequences by length, the offset should be 1.) - David Callan, Oct 31 2020

			The 7 tournament sequences of length 4 are 1234, 1235, 1236, 1245, 1246, 1247, 1248.

Alois P. Heinz, Table of n, a(n) for n = 0..85 (first 31 terms from T. D. Noe)
M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.
E. Neuwirth, Computing tournament sequence numbers efficiently..., Séminaire Lotharingien de Combinatoire, B47h (2002), 12 pp.
Mauro Torelli, Increasing integer sequences and Goldbach's conjecture, RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, 40:2 (2006), pp. 107-121.
Index entries for sequences related to tournaments

Cf. A058222, A058223, A093729, A093655.

Forms column 0 of triangle A097710.

t[n_?Negative, ] = 0; t[0, ] = 1; t[, 0] = 0; t[n, k_] /; k <= n :=  t[n, k] = t[n, k-1] - t[n-1, k] + t[n-1, 2k-1] + t[n-1, 2 k]; t[n_, k_] /; k > n :=  t[n, k] =Sum[(-1)^(j-1) Binomial[n+1, j]*t[n, k-j] , {j, 1, n+1}]; Table[t[n, 1], {n, 0, 15} ] (* Jean-François Alcover, May 17 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))))))} /*(Cook-Kleber)*/ a(n)=T(n,1)

@CachedFunction
def T(n, k):
    if n<0: return 0
    elif n==0: return 1
    elif k==0: return 0
    elif kA008934(n): return T(n,1)
[A008934(n) for n in range(31)] # G. C. Greubel, Feb 22 2024

From Paul D. Hanna, Apr 14 2004: (Start)
a(n) = A093729(n, 1).
a(n) = A093655(2^n). (End)
a(n) = A097710(n, 0). - Paul D. Hanna, Aug 24 2004
From Benedict W. J. Irwin, Nov 26 2016: (Start)
Conjecture: a(n) is given by a series of nested sums as follows:
  a(2) = Sum_{i=1..2} 1,
  a(3) = Sum_{i=1..2} Sum_{j=1..i+2} 1,
  a(4) = Sum_{i=1..2} Sum_{j=1..i+2} Sum_{k=1..i+j+2} 1,
  a(5) = Sum_{i=1..2} Sum_{j=1..i+2} Sum_{k=1..i+j+2} Sum_{l=1..i+j+k+2} 1.
(End)

A097710 Lower triangular matrix T, read by rows, such that row (n) is formed from the sums of adjacent terms in row (n-1) of the matrix square T^2, with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 7, 13, 7, 1, 41, 88, 61, 15, 1, 397, 951, 781, 257, 31, 1, 6377, 16691, 15566, 6231, 1041, 63, 1, 171886, 484490, 500057, 231721, 48303, 4161, 127, 1, 7892642, 23701698, 26604323, 13843968, 3406505, 374127, 16577, 255, 1

Offset: 0

Paul D. Hanna, Aug 22 2004

Column 0 is equal to sequence A008934, which is the number of tournament sequences.

This triangle has the same row sums and first column terms as in rows 2^n, for n>=0, of triangle A093654.

			Rows of this triangle T begin:
       1;
       1,      1;
       2,      3,      1;
       7,     13,      7,      1;
      41,     88,     61,     15,     1;
     397,    951,    781,    257,    31,    1;
    6377,  16691,  15566,   6231,  1041,   63,   1;
  171886, 484490, 500057, 231721, 48303, 4161, 127, 1;
Rows of T^2 begin:
        1;
        2,        1;
        7,        6,        1;
       41,       47,       14,       1;
      397,      554,      227,      30,      1;
     6377,    10314,     5252,     979,     62,     1;
   171886,   312604,   187453,   44268,   4035,   126,   1;
  7892642, 15809056, 10795267, 3048701, 357804, 16323, 254, 1;
The sums of adjacent terms in row (n) of T^2 forms row (n+1) of T:
  T(5,0) = T^2(4,0) = 397;
  T(5,1) = T^2(4,0) + T^2(4,1) = 397 + 554 = 951;
  T(5,2) = T^2(4,1) + T^2(4,2) = 554 + 227 = 781.
Rows of matrix inverse T^(-1) begins:
   1;
  -1,     1;
   1,    -3,      1;
  -1,     8,     -7,     1;
   1,   -25,     44,   -15,      1;
  -1,   111,   -346,   208,    -31,    1;
   1,  -809,   4045, -3720,    912,  -63,    1;
  -1, 10360, -77351, 99776, -35136, 3840, -127, 1; ...
which is a signed version of A097712.

Paul D. Hanna, Table of n, a(n) for n = 0..495, of rows 0..30 of triangle in flattened form.

Cf. A000007, A000225, A093654, A097712.

Cf. A008934 (column k=0), A093657 (row sums), A097711 (column k=1).

T[n_, k_] := T[n, k] = Which[n<0 || k>n, 0, n == k, 1, k == 0, Sum[T[n-1, j]*T[j, 0], {j, 0, n-1}], True, Sum[T[n-1, j]*T[j, k-1], {j, 0, n-1}] + Sum[T[n-1, j]*T[j, k], {j, 0, n-1}]]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 23 2016, adapted from PARI *)

/* Using Recurrence relation: */
{T(n,k) = if(n<0||k>n, 0, if(n==k,1, if(k==0, sum(j=0,n-1, T(n-1,j)*T(j,0)),  sum(j=0,n-1, T(n-1,j)*T(j,k-1)) + sum(j=0,n-1, T(n-1,j)*T(j,k));)))}
for(n=0,8, for(k=0,n, print1(T(n,k),", "));print(""))

/* Faster: using Matrix generating method: */
{T(n,k) = my(M=matrix(2,2,r,c,if(r>=c,1))); for(i=1,n,
N=matrix(#M+1,#M+1,r,c, if(r>=c, if(r<=#M,M[r,c], if(c>1,(M^2)[r-1,c-1]) + if(c<=#M,(M^2)[r-1,c])) ));
M=N;); M[n+1,k+1]}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print("")) \\ Paul D. Hanna, Nov 27 2016

@CachedFunction
def T(n, k): # T = A097710
    if n< 0 or k<0 or k>n: return 0
    elif k==n: return 1
    elif k==0: return sum(T(n-1,j)*T(j,0) for j in range(n))
    else: return sum(T(n-1, j)*(T(j, k-1)+T(j,k)) for j in range(n))
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 21 2024

T(n, k) = T^2(n-1, k-1) + T^2(n-1, k) for n>=1 and k>1, with T(n, 1) = T^2(n-1, 1) and T(n,n) = 1 for n>=0, where T^2 is the matrix square of this triangle T.
T(n, k) = Sum_{j=0..n-1} T(n-1, j)*(T(j, k-1) + T(j,k)), with T(n, 0) = Sum_{j=0..n-1} T(n-1,j)*T(j,0), and T(n, n) = 1.
T(n, 0) = A008934(n).
T(n, 1) = A097711(n).
Sum_{k=0..n} T(n, k) = A093657(n+1) (row sums).
From G. C. Greubel, Feb 21 2024: (Start)
T(n, n-1) = A000225(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)

cp's OEIS Frontend

A093655 First column of lower triangular matrix A093654.

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A093657 2^(n-1)-th term of the row sums of triangle A093654.

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A093656 Row sums of lower triangular matrix A093654.

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A008934 Number of tournament sequences: sequences (a_1, a_2, ..., a_n) with a_1 = 1 such that a_i < a_{i+1} <= 2*a_i for all i.

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A097710 Lower triangular matrix T, read by rows, such that row (n) is formed from the sums of adjacent terms in row (n-1) of the matrix square T^2, with T(0,0)=1.

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