A093730
Antidiagonal sums of triangle A093729, which enumerates the number of nodes in the tree of tournament sequences.
Original entry on oeis.org
1, 1, 2, 5, 18, 102, 949, 14731, 386060, 17323052, 1351157580, 185867701560, 45682244004244, 20283964291276804, 16423005586691362832, 24434416299840231799694, 67236458264587977465709983
Offset: 0
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T[n_, k_] := 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[(-1)^(j-1) * Binomial[n+1, j]*T[n, k-j], {j, 1, n+1}]]]]]; a[n_] := Sum[T[n-k, k], {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Oct 06 2016, translated from PARI *)
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{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)=sum(k=0,n,T(n-k,k))
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@CachedFunction
def T(n, k): # T = A093729
if n<0: return 0
elif n==0: return 1
elif k==0: return 0
elif kA093730(n): return sum(T(n-k,k) for k in range(n+1))
[A093730(n) for n in range(31)] # G. C. Greubel, Feb 22 2024
A097711
Column 1 of triangle A097710, in which row (n) is formed from the sums of the adjacent terms in row (n-1) of the matrix square of A097710.
Original entry on oeis.org
1, 3, 13, 88, 951, 16691, 484490, 23701698, 1990327810, 291750344191, 75757923092106, 35286335933354828, 29791358931890967248, 45989706937220594708463, 130760311958838053647976497
Offset: 0
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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 *)
A097711[n_]:= T[n+1,1];
Table[A097711[n], {n,0,30}] (* G. C. Greubel, Feb 21 2024 *)
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@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 A097711(n): return T(n+1,1)
[A097711(n) for n in range(31)] # G. C. Greubel, Feb 21 2024
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
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,...
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{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])}
A137273
Number of partitions of n-th Fibonacci number into Fibonacci parts obtained by iteratively dividing F(k) into F(n-1) and F(n-2); number of sub-Fibonacci sequences of length n starting with 1,0.
Original entry on oeis.org
1, 1, 2, 3, 6, 13, 37, 134, 659, 4416, 41343, 546577, 10345970, 283128770, 11306821624, 664047579721, 57753201767477, 7483309752358051
Offset: 1
For the sub-Fibonacci sequence 1,0,1,1,1,2, we split F(6)=8 into 5,3; split the 5 into 3,2; split one 3 into 2,1; and split both 2's into 1,1. This gives the partition [3,1^5].
[2^4] is the smallest partition of a Fibonacci number into Fibonacci parts that cannot be obtained in this way.
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nextfibpart(m) = local(s); s=matsize(m);matrix(s[2],s[1]+s[2]-1,i,j,sum(k=max(j-i+1,1),s[1],m[k,i]))
alist(n) = {local(v,m); v=vector(n,j,1); m=[0;1]; for(i=3,n, m=nextfibpart(m);v[i]=sum(j=1,matsize(m)[1],sum(k=1,matsize(m)[2],m[j,k]))); v}
A355129
a(n) is the number of integer sequences b(0..n) of length n+1, with 0 <= b(k) <= k! and monotonic b(k) <= b(k+1).
Original entry on oeis.org
2, 3, 7, 40, 856, 91821, 60080136, 279276911843, 10503211888973754, 3585680755683196123365, 12323227994417456429490342865, 468378989392773003347310901356953089, 214565221409985003242070442557341938941878313, 1282499669290042152350268651085002913530161723080398635
Offset: 0
For a(0) we get two possible sequences:
{0}, {1}.
For a(1) we get three possible sequences:
{0, 0}, {0, 1}, {1, 1}.
For a(2) = 7 we get:
{0, 0, 0}, {0, 0, 1}, {0, 0, 2}, {0, 1, 1},
{0, 1, 2}, {1, 1, 1}, {1, 1, 2}.
Cf.
A000108 (if we change the definition into 0 <= b(k) <= k).
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a(n) = binomial((n-1)! + n-1, n-1) + binomial((n-1)! + n-2, n-1) + sum(r = 1, n-2, sum(k = 0, r-1 ,binomial((n-1)! - r! - k+n - 2, n-1)*binomial(r-1,k)*a(r)*(-1)^(k+1)))
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