keyword:eigen - cp's OEIS frontend

A032200 Number of rooted compound windmills (mobiles) of n nodes.

1, 1, 2, 4, 9, 20, 51, 128, 345, 940, 2632, 7450, 21434, 62174, 182146, 537369, 1596133, 4767379, 14312919, 43162856, 130695821, 397184252, 1211057426, 3703794849, 11358759346, 34923477315, 107627138308, 332404636811

Offset: 1

Christian G. Bower

Also the number of locally necklace plane trees with n nodes, where a plane tree is locally necklace if the sequence of branches directly under any given node is lexicographically minimal among its cyclic permutations. - Gus Wiseman, Sep 05 2018

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(5) = 9 locally necklace plane trees:
  ((((o))))
  (((oo)))
  ((o(o)))
  (o((o)))
  ((o)(o))
  ((ooo))
  (o(oo))
  (oo(o))
  (oooo)
(End)

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.84).

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)
Index entries for sequences related to mobiles

Cf. A029768, A038037, A055340.

Cf. A000108, A007853, A032171, A254040, A304173, A304175, A317852.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
neckplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[neckplane/@c],neckQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[neckplane[n]],{n,10}] (* Gus Wiseman, Sep 05 2018 *)

CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=O(1));for(i=1, n, p=1+CIK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

Shifts left under "CIK" (necklace, indistinct, unlabeled) transform.

A032347 Inverse binomial transform of A032346.

Original entry on oeis.org

1, 0, 1, 2, 6, 21, 82, 354, 1671, 8536, 46814, 273907, 1700828, 11158746, 77057021, 558234902, 4230337018, 33448622893, 275322101318, 2354401779494, 20878592918183, 191682453823420, 1819147694792802

Offset: 0

Joe K. Crump (joecr(AT)carolina.rr.com)

Branko Dragovich, On Summation of p-Adic Series, arXiv:1702.02569 [math.NT], 2017.
N. J. A. Sloane, Transforms

Cf. A032346, A046934.

a[0] = 1; a[1] = 0; a[n_] := a[n] = 1 + Sum[Binomial[n-1, j]*a[j], {j, 2, n-1}]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Oct 08 2013, after Jon Perry *)
nmax = 20; Assuming[x > 0, CoefficientList[Series[E^(E^x) * (1/E + ExpIntegralEi[-1] - ExpIntegralEi[-E^x]), {x, 0, nmax}], x] ] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 10 2020 *)

{a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=1 - x * (1 - subst(A, x, x/(1-x)) / (1 - x))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Jul 10 2020

E.g.f. satisfies A' = exp(x) A - 1.
Recurrence: a(1)=0, a(2)=1, for n > 2, a(n) = 1 + Sum_{j=2..n-1} binomial(n-1, j)*a(j). - Jon Perry, Apr 26 2005
G.f. A(x) satisfies: A(x) = 1 - x * (1 - A(x/(1 - x)) / (1 - x)). - Ilya Gutkovskiy, Jul 10 2020

A035357 Number of increasing asymmetric rooted polygonal cacti with bridges (mixed Husimi trees).

Original entry on oeis.org

1, 1, 1, 7, 39, 409, 4687, 62822, 945250, 15999616, 300150210, 6198330586, 139779046596, 3420083177362, 90241503643208, 2554721759776914, 77240614583288344, 2484170781778551036

Offset: 1

Christian G. Bower, Nov 15 1998

Cf. A000083, A000237, A000314, A035082, A035349-A035356.

Shifts left under transform T where Ta = EGJ(BHJ(a)).

A038037 Number of labeled rooted compound windmills (mobiles) with n nodes.

Original entry on oeis.org

1, 2, 9, 68, 730, 10164, 173838, 3524688, 82627200, 2198295360, 65431163160, 2154106470240, 77714083773456, 3048821300491680, 129221979665461200, 5884296038166954240, 286492923374605966080, 14851359950834255500800

Offset: 1

Christian G. Bower, Sep 15 1998

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.83).

Alois P. Heinz, Table of n, a(n) for n = 1..140
C. G. Bower, Transforms (2)
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 454
B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
Index entries for sequences related to mobiles

Cf. A029768, A032200, A055349.

logtr:= proc(p) local b; b:=proc(n) option remember; local k; if n=0 then 1 else p(n)- add(k *binomial(n,k) *p(n-k) *b(k), k=1..n-1)/n fi end end: b:= logtr(-a): a:= n-> `if`(n<=1,1, -n*b(n-1)): seq(a(n), n=1..25); # Alois P. Heinz, Sep 14 2008

a[n_] = Sum[Binomial[n, j]*Abs[StirlingS1[n-1, j]]*j!, {j, 0, n}]; Array[a, 18]
(* Jean-François Alcover, Jun 22 2011, after Vladimir Kruchinin *)

Vec(serlaplace(serreverse(x/(1 - log(1-x + O(x^20)))))) \\ Andrew Howroyd, Sep 19 2018

Divides by n and shifts left under "CIJ" (necklace, indistinct, labeled) transform.
E.g.f. A(x) satisfies A(x) = x-x*log(1-A(x)). [Corrected by Andrey Zabolotskiy, Sep 16 2022]
a(n) = Sum_{j=0..n} binomial(n,j)*abs(Stirling1(n-1,j))*j!, n > 0. - Vladimir Kruchinin, Feb 03 2011
a(n) ~ sqrt(-1-LambertW(-1,-exp(-2))) * (-LambertW(-1,-exp(-2)))^(n-1) * n^(n-1) / exp(n). - Vaclav Kotesovec, Dec 27 2013
E.g.f.: series reversion of x/(1 - log(1-x)). - Andrew Howroyd, Sep 19 2018

A038046 Shifts left under transform T where Ta is (identity) DCONV a.

Original entry on oeis.org

1, 1, 3, 6, 12, 17, 32, 39, 63, 81, 120, 131, 213, 226, 311, 377, 503, 520, 742, 761, 1031, 1169, 1442, 1465, 2008, 2093, 2558, 2801, 3465, 3494, 4591, 4622, 5628, 6054, 7111, 7390, 9321, 9358, 10899, 11616, 13873, 13914, 17070, 17113, 20063, 21509, 24462

Offset: 1

Christian G. Bower

Eigensequence of triangle A126988. (i.e. the sequence shifts upon multiplication from the left by triangle A126988). - Gary W. Adamson, Apr 27 2009

Number of planted achiral trees with a distinguished leaf. - Gus Wiseman, Jul 31 2018

			From _Gus Wiseman_, Jul 31 2018: (Start)
The a(5) = 12 planted achiral trees with a distinguished leaf:
  (Oooo), (oOoo), (ooOo), (oooO),
  ((O)(o)), ((o)(O)),
  ((Ooo)), ((oOo)), ((ooO)),
  (((Oo))), (((oO))),
  ((((O)))).
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A126988. - Gary W. Adamson, Apr 27 2009

Cf. A000081, A002033, A003238, A004111, A007554, A038046, A067824, A317580.

a:= proc(n) option remember; `if`(n<2, n, (m-> m*
      add(a(d)/d, d=numtheory[divisors](m)))(n-1))
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, May 09 2019

a[n_]:=If[n==1,1,Sum[d*a[(n-1)/d],{d,Divisors[n-1]}]];
Array[a,30] (* Gus Wiseman, Jul 31 2018 *)

a(1) = 1; a(n > 1) = Sum_{d|(n-1)} d * a((n-1)/d). - Gus Wiseman, Jul 31 2018

G.f. A(x) satisfies: A(x) = x * (1 + Sum_{j>=1} j*A(x^j)). - Ilya Gutkovskiy, May 09 2019

A051163 Sequence is defined by property that (a0,a1,a2,a3,...) = binomial transform of (a0,a0,a1,a1,a2,a2,a3,a3,...).

Original entry on oeis.org

1, 2, 5, 12, 30, 76, 194, 496, 1269, 3250, 8337, 21428, 55184, 142376, 367916, 952000, 2466014, 6393372, 16586678, 43054344, 111801908, 290412296, 754543052, 1960808160, 5096293794, 13247503540, 34440553562, 89549255592, 232868582328, 605646682144

Offset: 0

Jonas Wallgren

Equals the self-convolution of A027826. Also equals antidiagonal sums of symmetric square array A100936. - Paul D. Hanna, Nov 22 2004

Equals eigensequence of triangle A152198. - Gary W. Adamson, Nov 28 2008

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

Cf. A051164, A051165, A051166, A027826, A100936, A100937, A152198.

a:= proc(n) option remember; add(`if`(k<2, 1,
      a(iquo(k, 2)))*binomial(n, k), k=0..n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 08 2015

a[n_] := a[n] = 1 + Sum[Binomial[k, j]*Binomial[n-k, j]*a[j], {k, 1, n}, {j, 0, n-k}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 11 2015 *)

a(n)=1+sum(k=1,n,sum(j=0,n-k,binomial(k,j)*binomial(n-k,j)*a(j)))

a(n)=local(A,m); if(n<0,0,m=1; A=1+O(x); while(m<=n,m*=2; A=subst(A,x,(x/(1-x))^2)/(1-x)); polcoeff(A^2,n))
for(n=0,40,print1(a(n),", ")) \\ Paul D. Hanna, Nov 22 2004

a(n) = 1 + Sum_{k=1..n} Sum_{j=0..n-k} C(k, j)*C(n-k, j)*a(j). - Paul D. Hanna, Nov 22 2004
G.f. A(x) satisfies: A(x) = A(x^2/(1-x)^2)/(1-x)^2 and A(x^2) = A(x/(1+x))/(1+x)^2. - Paul D. Hanna, Nov 22 2004
a(0) = 1; a(n) = Sum_{k=0..floor(n/2)} binomial(n+1,2*k+1) * a(k). - Ilya Gutkovskiy, Apr 07 2022

More terms from Vladeta Jovovic, Jul 26 2002

A054090 Triangular array generated by its row sums: T(n,0) = 1 for n >= 0, T(n,1) = r(n-1), T(n,k) = T(n,k-1) - (-1)^k * r(n-k) for k = 2, 3, ..., n, n >= 2, r(h) = sum of the numbers in row h of T.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 2, 3, 1, 10, 6, 8, 7, 1, 32, 22, 26, 24, 25, 1, 130, 98, 108, 104, 106, 105, 1, 652, 522, 554, 544, 548, 546, 547, 1, 3914, 3262, 3392, 3360, 3370, 3366, 3368, 3367, 1, 27400, 23486, 24138, 24008, 24040, 24030, 24034, 24032, 24033

Offset: 0

Clark Kimberling

			Triangle begins as:
  1;
  1,     1;
  1,     2,     1;
  1,     4,     2,     3;
  1,    10,     6,     8,     7;
  1,    32,    22,    26,    24,    25;
  1,   130,    98,   108,   104,   106,   105;
  1,   652,   522,   554,   544,   548,   546,   547;
  1,  3914,  3262,  3392,  3360,  3370,  3366,  3368,  3367;
  1, 27400, 23486, 24138, 24008, 24040, 24030, 24034, 24032, 24033;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A054091 (row sums).

T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==1, Sum[T[n-1,j], {j,0,n-1}], T[n,k-1] - (-1)^k*Sum[T[n-k,j], {j,0,n-k}]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 23 2022 *)

{T(n, k)= local(A); if(k<0||k>n, 0, if(k==0, 1, A=vector(n, i, (i>1)+1); for(i=2, n-1, A[i+1]=(i-1)*A[i]+2); sum(i=0, k-1, (-1)^i*A[n-i])))} /* Michael Somos, Nov 19 2006 */

@CachedFunction
def T(n, k): # T = A054090
    if (k==0): return 1
    elif (k==1): return sum(T(n-1, j) for j in (0..n-1))
    else: return T(n, k-1) - (-1)^k*sum(T(n-k, j) for j in (0..n-k))
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 23 2022

T(n, k) = T(n, k-1) - (-1)^k * Sum_{j=0..n-k} T(n-k, j), with T(n, 0) = 1, and T(n, 1) = Sum_{j=0..n-1} T(n-1, j).

Sum_{k=0..n} T(n, k) = A054091(n).

A055349 Triangle of labeled mobiles (circular rooted trees) with n nodes and k leaves.

Original entry on oeis.org

1, 2, 0, 6, 3, 0, 24, 36, 8, 0, 120, 360, 220, 30, 0, 720, 3600, 4200, 1500, 144, 0, 5040, 37800, 71400, 47250, 11508, 840, 0, 40320, 423360, 1176000, 1234800, 545664, 98784, 5760, 0, 362880, 5080320, 19474560, 29635200, 20469456, 6618528, 940896, 45360, 0

Offset: 1

Christian G. Bower, May 15 2000

			Triangle begins:
     1;
     2,     0;
     6,     3,     0;
    24,    36,     8,     0;
   120,   360,   220,    30,     0;
   720,  3600,  4200,  1500,   144,   0;
  5040, 37800, 71400, 47250, 11508, 840, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Index entries for sequences related to mobiles

Row sums give A038037.
Columns 1..8 are A000142, A055303, A055350, A055351, A055352, A055353, A055354, A055355.
Cf. A055340, A055356, A055363.

T[rows_] := {{1}}~Join~((cc = CoefficientList[#, y]; Append[Rest[cc], 0] * Length[cc]!)& /@ (CoefficientList[InverseSeries[x/(y-Log[1-x + O[x]^rows] ), x], x][[3;;]]));
T[9] // Flatten (* Jean-François Alcover, Oct 31 2019 *)

A(n)={my(v=Vec(serlaplace(serreverse(x/(y - log(1-x + O(x^n))))))); vector(#v, i, Vecrev(v[i]/y, i))}
{ my(T=A(10)); for(i=1, #T, print(T[i])) } \\ Andrew Howroyd, Sep 23 2018

E.g.f. satisfies A(x, y) = x*y - x*log(1-A(x, y)). [Corrected by Sean A. Irvine, Mar 19 2022]

A144018 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where sequence a_k of column k has a_k(0)=0, followed by (k+1)-fold 1 and a_k(n) shifts k places left under Euler transform.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 9, 3, 2, 1, 1, 20, 6, 3, 2, 1, 1, 48, 10, 5, 3, 2, 1, 1, 115, 20, 8, 5, 3, 2, 1, 1, 286, 36, 14, 7, 5, 3, 2, 1, 1, 719, 72, 23, 12, 7, 5, 3, 2, 1, 1, 1842, 137, 40, 18, 11, 7, 5, 3, 2, 1, 1, 4766, 275, 69, 30, 16, 11, 7, 5, 3, 2, 1, 1, 12486, 541, 121, 47, 25, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Alois P. Heinz, Sep 07 2008

			T(5,1) = ([1,2,4]*[1,1,4] + [1]*[1]*4 + [1,2]*[1,1]*2 + [1,3]*[1,2]*1)/4 = 36/4 = 9.
Triangle begins:
    1;
    1,  1;
    2,  1,  1;
    4,  2,  1,  1;
    9,  3,  2,  1, 1;
   20,  6,  3,  2, 1, 1;
   48, 10,  5,  3, 2, 1, 1;
  115, 20,  8,  5, 3, 2, 1, 1;
  286, 36, 14,  7, 5, 3, 2, 1, 1;
  719, 72, 23, 12, 7, 5, 3, 2, 1, 1;

Alois P. Heinz, Rows n = 1..141, 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]
N. J. A. Sloane, Transforms

Columns k=1..10 give A000081, A007562, A218020, A218021, A218022, A218023, A218024, A218025, A218026, A218027.
T(2n,n) gives A000041(n).
Cf. A316074.

etrk:= proc(p) proc(n, k) option remember; `if`(n=0, 1,
         add(add(d*p(d, k), d=numtheory[divisors](j))*
         procname(n-j, k), j=1..n)/n)
       end end:
B:= etrk(T):
T:= (n, k)-> `if`(n<=k, `if`(n=0, 0, 1), B(n-k, k)):
seq(seq(T(n, k), k=1..n), n=1..14);

etrk[p_] := Module[{f}, f[n_, k_] := f[n, k] = If[n == 0, 1, (Sum[Sum[d*p[d, k], {d, Divisors[j]}]*f[n-j, k], {j, 1, n-1}] + Sum[d*p[d, k], {d, Divisors[n]}])/n]; f]; b = etrk[t]; t[n_, k_] := If[n <= k, If[n == 0, 0, 1], b[n-k, k]]; Table[t[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 01 2013, after Alois P. Heinz *)

A000998 From a differential equation.

Original entry on oeis.org

1, 3, 6, 11, 24, 69, 227, 753, 2451, 8004, 27138, 97806, 375313, 1511868, 6292884, 26826701, 116994453, 523646202, 2414394601, 11487130362, 56341183365, 284110648983, 1468690344087, 7766823788295, 41976012524088, 231812530642644, 1308325741771908

Offset: 0

N. J. A. Sloane

When preceded by {0, 0, 1, 0, 0}, this sequence shifts 3 places under binomial transform. - Olivier Gérard, Aug 12 2016

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..695
S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.

b:= proc(n) option remember; `if`(n<3, [0$2, 1][n+1],
      add(binomial(n-3, j)*b(j), j=0..n-3))
    end:
a:= n-> b(n+5):
seq(a(n), n=0..30);  # Alois P. Heinz, May 21 2019

b[n_] := b[n] = If[n<3, {0, 0, 1}[[n+1]], Sum[Binomial[n-3, j] b[j], {j, 0, n-3}]];
a[n_] := b[n+5];
a /@ Range[0, 30] (* Jean-François Alcover, Oct 27 2020, after Alois P. Heinz *)

G.f.: A(x) = Sum(x^(3*k-3)/Product(1-l*x,l = 0 .. k)^3,k = 0 .. infinity). - Vladeta Jovovic, Feb 05 2008

More terms from Vladeta Jovovic, Feb 05 2008

cp's OEIS Frontend

A032200 Number of rooted compound windmills (mobiles) of n nodes.

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A032347 Inverse binomial transform of A032346.

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A035357 Number of increasing asymmetric rooted polygonal cacti with bridges (mixed Husimi trees).

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A038037 Number of labeled rooted compound windmills (mobiles) with n nodes.

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A038046 Shifts left under transform T where Ta is (identity) DCONV a.

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A051163 Sequence is defined by property that (a0,a1,a2,a3,...) = binomial transform of (a0,a0,a1,a1,a2,a2,a3,a3,...).

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A054090 Triangular array generated by its row sums: T(n,0) = 1 for n >= 0, T(n,1) = r(n-1), T(n,k) = T(n,k-1) - (-1)^k * r(n-k) for k = 2, 3, ..., n, n >= 2, r(h) = sum of the numbers in row h of T.

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