Original entry on oeis.org
1, 1, 1, 2, 4, 10, 26, 71, 197, 564, 1639, 4833, 14406, 43374, 131652, 402525, 1238419, 3831520, 11912913, 37204431, 116655147, 367100319, 1159026041, 3670339794, 11655070593, 37104257405, 118398974620, 378627600346, 1213247498254, 3894924465243
Offset: 0
-
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(b((i-1)$2), j)*g(n-i*j, i-1), j=0..n/i)))
end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(g(i$2), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> g(n, n):
seq(a(n), n=0..40); # Alois P. Heinz, May 20 2013
-
g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[b[i-1, i-1], j]* g[n-i*j, i-1], {j, 0, n/i}]]];
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*b[n- i*j, i-1], {j, 0, n/i}]]];
a[n_] := g[n, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 22 2017, after Alois P. Heinz *)
A316101
Sequence a_k of column k shifts left when Weigh transform is applied k times with a_k(n) = n for n<2; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 3, 3, 1, 0, 1, 1, 1, 4, 6, 6, 1, 0, 1, 1, 1, 5, 10, 16, 12, 1, 0, 1, 1, 1, 6, 15, 32, 43, 25, 1, 0, 1, 1, 1, 7, 21, 55, 105, 120, 52, 1, 0, 1, 1, 1, 8, 28, 86, 210, 356, 339, 113, 1, 0, 1, 1, 1, 9, 36, 126, 371, 826, 1227, 985, 247, 1
Offset: 0
Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, 8, 9, ...
1, 3, 6, 10, 15, 21, 28, 36, 45, ...
1, 6, 16, 32, 55, 86, 126, 176, 237, ...
1, 12, 43, 105, 210, 371, 602, 918, 1335, ...
1, 25, 120, 356, 826, 1647, 2961, 4936, 7767, ...
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Columns k=0-10 give:
A057427,
A004111,
A007561,
A316103,
A316104,
A316105,
A316106,
A316107,
A316108,
A316109,
A316110.
-
wtr:= proc(p) local b; b:= proc(n, i) option remember;
`if`(n=0, 1, `if`(i<1, 0, add(binomial(p(i), j)*
b(n-i*j, i-1), j=0..n/i))) end: j-> b(j$2)
end:
g:= proc(k) option remember; local b, t; b[0]:= j->
`if`(j<2, j, b[k](j-1)); for t to k do
b[t]:= wtr(b[t-1]) od: eval(b[0])
end:
A:= (n, k)-> g(k)(n):
seq(seq(A(n, d-n), n=0..d), d=0..14);
-
wtr[p_] := Module[{b}, b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[p[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]]; b[#, #]&];
g[k_] := g[k] = Module[{b, t}, b[0][j_] := If[j < 2, j, b[k][j - 1]]; For[ t = 1, t <= k + 1, t++, b[t] = wtr[b[t - 1]]]; b[0]];
A[n_, k_] := g[k][n];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jul 10 2018, after Alois P. Heinz *)
A035081
Number of increasing asymmetric rooted connected graphs where every block is a complete graph.
Original entry on oeis.org
1, 1, 1, 7, 27, 167, 1451, 12672, 133356, 1573608, 20731512, 299642958, 4732486932, 81201040470, 1500094187292, 29730606352920, 628968809015766, 14147458062941100, 337143091156288002, 8485143902146640124
Offset: 1
-
EGJ(v)={Vec(serlaplace(prod(k=1, #v, (1 + x^k/k! + O(x*x^#v))^v[k]))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, v=concat([1], EGJ(EGJ(v)))); v} \\ Andrew Howroyd, Sep 11 2018
A035080
Number of asymmetric connected graphs where every block is a complete graph.
Original entry on oeis.org
1, 1, 0, 0, 0, 0, 1, 3, 7, 21, 60, 168, 472, 1344, 3843, 11104, 32305, 94734, 279708, 831401, 2485877, 7474667, 22589771, 68594611, 209198103, 640591332, 1968920180, 6072766832, 18791062733, 58321579888, 181524367875, 566488767763, 1772261945866, 5557515157647
Offset: 0
-
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(b((i-1)$2), j)*g(n-i*j, i-1), j=0..n/i)))
end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(g(i$2), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b((n-1)$2)+g(n$2)-add(b((i-1)$2)*g((n-i)$2), i=0..n):
seq(a(n), n=0..40); # Alois P. Heinz, May 20 2013
-
g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[b[i-1, i-1], j]*g[n-i*j, i-1], {j, 0, n/i}]]]; b[n_, i_] := b[n, i] = If[n==0, 1, If[i < 1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n-1, n-1] + g[n, n] - Sum[b[i-1, i-1]*g[n-i, n-i], {i, 0, n}]; Table[ a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)
Showing 1-4 of 4 results.
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