A144042 -id:A144042 - cp's OEIS frontend

A007563 Number of rooted connected graphs where every block is a complete graph.

0, 1, 1, 3, 8, 25, 77, 258, 871, 3049, 10834, 39207, 143609, 532193, 1990163, 7503471, 28486071, 108809503, 417862340, 1612440612, 6248778642, 24309992576, 94905791606, 371691137827, 1459935388202, 5749666477454

Offset: 0

N. J. A. Sloane

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 71, (3.4.13).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1600 (first 200 terms from T. D. Noe)
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; 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]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 167
N. J. A. Sloane, Transforms

Cf. A007549, A030019, A035051, A035052, A035053.
Column k=2 of A144042.
Cf. A245566.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(a): c:= etr(b): a:= n-> if n=0 then 0 else c(n-1) fi: seq(a(n), n=0..25); # Alois P. Heinz, Sep 06 2008

etr[p_] := etr[p] = Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; a[0] = 0; a[n_] := etr[etr[a]][n-1]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 28 2013, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(v)))); concat([0], v)} \\ Andrew Howroyd, May 20 2018

Shifts left when Euler transform is applied twice.

a(n) ~ c * d^n / n^(3/2), where d = 4.189610958393826965527036454524044275... (see A245566), c = 0.1977574301782950818433893126632477845870281049591883888... . - Vaclav Kotesovec, Jul 26 2014

New description from Christian G. Bower, Oct 15 1998

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

Alois P. Heinz, Jun 24 2018

			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.
Rows include (offsets may differ): A000004, A000012, A000027, A000217, A134465.
Main diagonal gives A316102.
Cf. A144042, A316074.

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 *)

A305725 a(n) is the n-th term of the sequence that shifts left by one position when Euler transform is applied n times; a(0) = 0.

Original entry on oeis.org

0, 1, 1, 4, 19, 141, 1260, 14379, 192615, 2997844, 52869443, 1042233820, 22685350130, 540054008352, 13951013617182, 388499286052386, 11597298980187673, 369329248762633105, 12495176515326675179, 447453508295247538083, 16905177062269436114613

Offset: 0

Alois P. Heinz, Jun 21 2018

Alois P. Heinz, Table of n, a(n) for n = 0..200

Main diagonal of A144042.

Cf. A290354, A316102.

a(n) = A144042(n,n) for n>0, a(0) = 0.

A144035 Shifts left when Euler transform applied 3 times.

Original entry on oeis.org

0, 1, 1, 4, 13, 51, 197, 828, 3526, 15538, 69627, 317564, 1466868, 6853320, 32317354, 153636769, 735493374, 3542610091, 17155811156, 83480667707, 407969449010, 2001479974330, 9853652529076, 48666276924852, 241059431701113, 1197237945360797, 5960804381552300

Offset: 0

Alois P. Heinz, Sep 07 2008

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

3rd column of A144042.

Cf. A316103.

b:= ((proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1,
        add(add(d*p(d), d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
      end end)@@3)(a):
a:= n-> b(n-1):
seq(a(n), n=0..25);  # revised Alois P. Heinz, Aug 27 2018

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := Module[{a, b, t}, b[1] = etr[a]; For[ t = 2, t <= k, t++, b[t] = etr[b[t-1]]]; a = Function[m, If[m == 1, 1, b[k][m-1]]]; a[n]]; Table[ A[n, 3], {n, 0, 30} ] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

A144036 Shifts left when Euler transform applied 4 times.

Original entry on oeis.org

0, 1, 1, 5, 19, 89, 410, 2052, 10440, 54874, 293549, 1597621, 8807766, 49107289, 276358791, 1567866228, 8957204966, 51486464912, 297548288251, 1727856600935, 10076859047404, 58996263573440, 346614270372761, 2042929868812385, 12076076910981403

Offset: 0

Alois P. Heinz, Sep 07 2008

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

4th column of A144042.

Cf. A316104.

k:=4: with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:='a': b[1]:=etr(a): for t from 2 to k do b[t]:= etr(b[t-1]) od: a:= n-> `if`(n<2,n,b[k](n-1)): seq(a(n), n=0..30);

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := Module[{a, b, t}, b[1] = etr[a]; For[ t = 2, t <= k, t++, b[t] = etr[b[t-1]]]; a = Function[m, If[m == 1, 1, b[k][m-1]]]; a[n]]; Table[A[n, 4], {n, 0, 30} ] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

A144037 Shifts left when Euler transform applied 5 times.

Original entry on oeis.org

0, 1, 1, 6, 26, 141, 751, 4337, 25512, 154839, 956877, 6014098, 38292603, 246550449, 1602304287, 10497348917, 69253008919, 459673679526, 3067621127220, 20570132531365, 138528107124822, 936524246563021, 6353624897677005, 43242243814689566, 295161330408772329

Offset: 0

Alois P. Heinz, Sep 07 2008

Alois P. Heinz, Table of n, a(n) for n = 0..400
N. J. A. Sloane, Transforms

5th column of A144042.

Cf. A316105.

k:=5: with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:='a': b[1]:=etr(a): for t from 2 to k do b[t]:= etr(b[t-1]) od: a:= n-> `if`(n<2,n,b[k](n-1)): seq(a(n), n=0..30);

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := Module[{a, b, t}, b[1] = etr[a]; For[ t = 2, t <= k, t++, b[t] = etr[b[t-1]]]; a = Function[m, If[m == 1, 1, b[k][m - 1]]]; a[n]]; Table[A[n, 5], {n, 0, 30} ] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

A144038 Shifts left when Euler transform applied 6 times.

Original entry on oeis.org

0, 1, 1, 7, 34, 209, 1260, 8219, 54677, 374904, 2617996, 18588586, 133708072, 972484206, 7139186644, 52832157095, 393700752053, 2951755132720, 22250074031893, 168524281244364, 1281906305775734, 9788771511218660, 75010092629043843, 576625427728255451

Offset: 0

Alois P. Heinz, Sep 07 2008

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

6th column of A144042.

Cf. A316106.

k:=6: with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:='a': b[1]:=etr(a): for t from 2 to k do b[t]:= etr(b[t-1]) od: a:= n-> `if`(n<2,n,b[k](n-1)): seq(a(n), n=0..30);

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := Module[{a, b, t}, b[1] = etr[a]; For[ t = 2, t <= k, t++, b[t] = etr[b[t-1]]]; a = Function[m, If[m == 1, 1, b[k][m - 1]]]; a[n]]; Table[A[n, 6], {n, 0, 30} ] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

A144039 Shifts left when Euler transform applied 7 times.

Original entry on oeis.org

0, 1, 1, 8, 43, 295, 1982, 14379, 106464, 811769, 6304231, 49770106, 398042059, 3218655532, 26269422252, 216121646661, 1790428656802, 14922986525284, 125050910828179, 1052916751167676, 8903502888220215, 75579379572985970, 643818145873489913

Offset: 0

Alois P. Heinz, Sep 07 2008

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

7th column of A144042.

Cf. A316107.

k:=7: with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:='a': b[1]:=etr(a): for t from 2 to k do b[t]:= etr(b[t-1]) od: a:= n-> `if`(n<2,n,b[k](n-1)): seq(a(n), n=0..30);

k = 7; etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; b[1] = etr[a]; For[t = 2, t <= k, t++, b[t] = etr[b[t-1]]]; a[n_] := If[n<2, n, b[k][n-1]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

A144040 Shifts left when Euler transform applied 8 times.

Original entry on oeis.org

0, 1, 1, 9, 53, 401, 2967, 23659, 192615, 1613687, 13769538, 119419761, 1049154071, 9318828138, 83541286988, 754920397795, 6869170162829, 62884110339333, 578768613062948, 5352307127138701, 49708821972158231, 463445557084360740, 4335898935627480699

Offset: 0

Alois P. Heinz, Sep 07 2008

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

8th column of A144042.

Cf. A316108.

k:=8: with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:='a': b[1]:=etr(a): for t from 2 to k do b[t]:= etr(b[t-1]) od: a:= n-> `if`(n<2,n,b[k](n-1)): seq(a(n), n=0..30);

k = 8; etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; b[1] = etr[a]; For[t = 2, t <= k, t++, b[t] = etr[b[t-1]]]; a[n_] := If[n<2, n, b[k][n-1]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

A144041 Shifts left when Euler transform applied 9 times.

Original entry on oeis.org

0, 1, 1, 10, 64, 529, 4270, 37078, 328765, 2997844, 27840748, 262746590, 2511753957, 24274552279, 236771528856, 2327857190814, 23045073461778, 229523320443974, 2298252776303916, 23122596000124215, 233629865440550370, 2369682121825700424, 24119295413989862382

Offset: 0

Alois P. Heinz, Sep 07 2008

Alois P. Heinz, Table of n, a(n) for n = 0..960
N. J. A. Sloane, Transforms

9th column of A144042.

Cf. A316109.

k:=9: with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:='a': b[1]:=etr(a): for t from 2 to k do b[t]:= etr(b[t-1]) od: a:= n-> `if`(n<2,n,b[k](n-1)): seq(a(n), n=0..30);

k = 9; etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; b[1] = etr[a]; For[t = 2, t <= k, t++, b[t] = etr[b[t-1]]]; a[n_] := If[n<2, n, b[k][n-1]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

cp's OEIS Frontend

A007563 Number of rooted connected graphs where every block is a complete graph.

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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.

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A305725 a(n) is the n-th term of the sequence that shifts left by one position when Euler transform is applied n times; a(0) = 0.

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A144035 Shifts left when Euler transform applied 3 times.

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A144036 Shifts left when Euler transform applied 4 times.

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A144037 Shifts left when Euler transform applied 5 times.

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A144038 Shifts left when Euler transform applied 6 times.

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A144039 Shifts left when Euler transform applied 7 times.

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A144040 Shifts left when Euler transform applied 8 times.

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