A265202 -id:A265202 - cp's OEIS frontend

A255047 1 together with the positive terms of A000225.

1, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295

Offset: 0

Omar E. Pol, Feb 15 2015

Also, right border of A246674 arranged as an irregular triangle.
Essentially the same as A168604, A126646 and A000225.
Total number of lambda-parking functions induced by all partitions of n. a(0)=1: [], a(1)=1: [1], a(2)=3: [1], [2], [1,1], a(4)=7: [1], [2], [3], [1,1], [1,2], [2,1], [1,1,1]. - Alois P. Heinz, Dec 04 2015
Also, the decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 19 2017
Also number of multiset partitions of {1,1} U [n] into exactly 2 nonempty parts.  a(2) = 3: 111|2, 11|12, 1|112. - Alois P. Heinz, Aug 18 2017
Also, the number of unlabeled connected P-series (equivalently, connected P-graphs) with n+1 elements. - Salah Uddin Mohammad, Nov 19 2021

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
R. Stanley, Parking Functions, 2011.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A011782, A028310, A246674, A253909, A265007, A265202, A349276.
Row n=1 of A263159.
Column k=2 of A291117.
Cf. A078485.

[1] cat [2^n -1: n in [1..40]]; // G. C. Greubel, Feb 07 2021

CoefficientList[Series[(1 -2*x +2*x^2)/((1-x)*(1-2*x)), {x, 0, 33}], x] (* or *) LinearRecurrence[{3, -2}, {1,1,3}, 40] (* Vincenzo Librandi, Jul 20 2017 *)
Table[2^n -1 +Boole[n==0], {n, 0, 40}] (* G. C. Greubel, Feb 07 2021 *)

def A255047(n): return -1^(-1<Chai Wah Wu, Dec 21 2022

[1]+[2^n -1 for n in (1..40)] # G. C. Greubel, Feb 07 2021

From Alois P. Heinz, Feb 19 2015: (Start)
O.g.f.: (1 -2*x +2*x^2)/((1-x)*(1-2*x)).
E.g.f.: exp(2*x) - exp(x) + 1. (End)
a(n) = A078485(n+1) for n > 2. - Georg Fischer, Oct 22 2018

A265016 Total sum of number of lambda-parking functions, where lambda ranges over all partitions of n into distinct parts.

Original entry on oeis.org

1, 1, 2, 6, 9, 20, 43, 74, 130, 241, 493, 774, 1413, 2286, 3987, 7287, 11650, 19235, 31581, 50852, 80867, 141615, 214538, 349179, 541603, 859759, 1303221, 2054700, 3277493, 4960397, 7652897, 11662457, 17703655, 26603187, 40043433, 59384901, 92234897, 134538472

Offset: 0

Alois P. Heinz, Nov 30 2015

nonn

			The number of lambda-parking functions induced by the partitions of 4 into distinct parts:
5 by [1,3]: [1,1], [1,2], [2,1], [1,3], [3,1],
4 by [4]: [1], [2], [3], [4].
a(4) = 5 + 4 = 9.

Alois P. Heinz, Table of n, a(n) for n = 0..100
Richard P. Stanley, Parking Functions, 2011.

Row sums of A265017, A265018, A265019, A265020.

Cf. A000009, A265007, A265202.

p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
         -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
g:= (n, i, l)->  `if`(i*(i+1)/2n, 0, g(n-i, i-1, [i, l[]])))):
a:= n-> g(n$2, []):
seq(a(n), n=0..35);

p[l_] := With[{n = Length[l]}, n!*Det[Table[Function[t,
     If[t < 0, 0, l[[i]]^t/t!]][j - i + 1], {i, n}, {j, n}]]];
g[n_, i_, l_] := If[i (i + 1)/2 < n, 0, If[n == 0, p[l],
     g[n, i - 1, l] + If[i > n, 0, g[n - i, i - 1, Prepend[l, i]]]]];
a[n_] := If[n == 0, 1, g[n, n, {}]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Aug 20 2021, after Alois P. Heinz *)

A265208 Total number T(n,k) of lambda-parking functions induced by all partitions of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 3, 0, 4, 5, 0, 5, 10, 0, 6, 14, 16, 0, 7, 21, 25, 0, 8, 27, 43, 0, 9, 36, 74, 0, 10, 44, 107, 125, 0, 11, 55, 146, 189, 0, 12, 65, 207, 307, 0, 13, 78, 267, 471, 0, 14, 90, 342, 786, 0, 15, 105, 436, 1058, 1296, 0, 16, 119, 538, 1490, 1921

Offset: 0

Alois P. Heinz, Dec 04 2015

Differs from A265020 first at T(5,2). See example.

			T(5,2) = 10: There are two partitions of 5 into 2 distinct parts: [2,3], [1,4]. Together they have 10 lambda-parking functions: [1,1], [1,2], [1,3], [1,4], [2,1], [2,2], [2,3], [3,1], [3,2], [4,1]. Here [1,1], [1,2], [1,3], [2,1], [3,1] are induced by both partitions. But they are counted only once.
T(6,1) = 6: [1], [2], [3], [4], [5], [6].
T(6,2) = 14: [1,1], [1,2], [1,3], [1,4], [1,5], [2,1], [2,2], [2,3], [2,4], [3,1], [3,2], [4,1], [4,2], [5,1].
T(6,3) = 16: [1,1,1], [1,1,2], [1,1,3], [1,2,1], [1,2,2], [1,2,3], [1,3,1], [1,3,2], [2,1,1], [2,1,2], [2,1,3], [2,2,1], [2,3,1], [3,1,1], [3,1,2], [3,2,1].
Triangle T(n,k) begins:
00 :  1;
01 :  0,  1;
02 :  0,  2;
03 :  0,  3,   3;
04 :  0,  4,   5;
05 :  0,  5,  10;
06 :  0,  6,  14,  16;
07 :  0,  7,  21,  25;
08 :  0,  8,  27,  43;
09 :  0,  9,  36,  74;
10 :  0, 10,  44, 107,  125;
11 :  0, 11,  55, 146,  189;
12 :  0, 12,  65, 207,  307;
13 :  0, 13,  78, 267,  471;
14 :  0, 14,  90, 342,  786;
15 :  0, 15, 105, 436, 1058, 1296;
16 :  0, 16, 119, 538, 1490, 1921;

Alois P. Heinz, Rows n = 0..400, flattened
R. Stanley, Parking Functions, 2011

Columns k=0-2 give: A000007, A000027, A176222(n+1).
Row sums give A265202.
Cf. A000217, A000272, A003056, A206735 (the same for general partitions), A265020, A265145.

b:= proc(p, g, n, i, t) option remember; `if`(g=0, 0, p!/g!*x^p)+
      `if`(n (p-> seq(coeff(p, x, i), i=0..degree(p)))(
        `if`(n=0, 1, b(0$2, n, 1$2))):
seq(T(n), n=0..25);

b[p_, g_, n_, i_, t_] := b[p, g, n, i, t] = If[g==0, 0, p!/g!*x^p] + If[nJean-François Alcover, Feb 02 2017, translated from Maple *)

T(A000217(n),n) = A000272(n+1).

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A255047 1 together with the positive terms of A000225.

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A265016 Total sum of number of lambda-parking functions, where lambda ranges over all partitions of n into distinct parts.

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A265208 Total number T(n,k) of lambda-parking functions induced by all partitions of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

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Keywords

Comments

Examples

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Maple

Mathematica

Formula