A227682 -id:A227682 - cp's OEIS frontend

A238349 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k parts p at position p (fixed points), n>=0, 0<=k<=n.

1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 3, 4, 1, 0, 0, 6, 7, 3, 0, 0, 0, 11, 16, 4, 1, 0, 0, 0, 22, 29, 12, 1, 0, 0, 0, 0, 42, 60, 23, 3, 0, 0, 0, 0, 0, 82, 120, 47, 7, 0, 0, 0, 0, 0, 0, 161, 238, 100, 12, 1, 0, 0, 0, 0, 0, 0, 316, 479, 198, 30, 1, 0, 0, 0, 0, 0, 0, 0, 624, 956, 404, 61, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1235, 1910, 818, 126, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 25 2014

T(n*(n+3)/2,n) = A227682(n).
From Vaclav Kotesovec, Sep 07 2014: (Start)
In general, column k is asymptotic to c(k) * 2^n. The constants c(k) numerically:
c(0) = 0.144394047543301210639449860964615390044455952420342... = A048651/2
c(1) = 0.231997216225445223894202367545783700531838988546098... = c(0)*A065442
c(2) = 0.104261929557371534733906196116707679501974368826074...
c(3) = 0.017956317806894073430249112172514186063327165575720...
c(4) = 0.001343254222922697613125145839110293324517874530073...
c(5) = 0.000046459767012163920051487037952792359225887287888...
c(6) = 0.000000768651747857094917953943327540619110335556499...
c(7) = 0.000000006200599904985793344094393321042983316604040...
c(8) = 0.000000000024656652167851516076173236693314090168122...
c(9) = 0.000000000000048633746319332356416193899916110113745...
c(10)= 0.000000000000000047750743608910618576944191079881479...
c(20)= 1.05217230403079700467566...*10^(-63)
For big k is c(k) ~ m * 2^(-k*(k+1)/2), where m = 1/(4*c(0)) = 1/(2*A048651) = 1.7313733097275318...
(End)

			Triangle starts:
00:  1,
01:  0, 1,
02:  1, 1, 0,
03:  2, 1, 1, 0,
04:  3, 4, 1, 0, 0,
05:  6, 7, 3, 0, 0, 0,
06:  11, 16, 4, 1, 0, 0, 0,
07:  22, 29, 12, 1, 0, 0, 0, 0,
08:  42, 60, 23, 3, 0, 0, 0, 0, 0,
09:  82, 120, 47, 7, 0, 0, 0, 0, 0, 0,
10:  161, 238, 100, 12, 1, 0, 0, 0, 0, 0, 0,
11:  316, 479, 198, 30, 1, 0, 0, 0, 0, 0, 0, 0,
12:  624, 956, 404, 61, 3, 0, 0, 0, 0, 0, 0, 0, 0,
13:  1235, 1910, 818, 126, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0,
14:  2449, 3817, 1652, 258, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
15:  4864, 7633, 3319, 537, 30, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
...
From _Gus Wiseman_, Apr 03 2022: (Start)
Row n = 5 counts the following compositions (empty columns indicated by dots):
  (5)     (14)     (113)   .  .  .
  (23)    (32)     (122)
  (41)    (131)    (1211)
  (212)   (221)
  (311)   (1112)
  (2111)  (1121)
          (11111)
(End)

M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10010 (rows 0..140, flattened)
M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.

Row sums are A011782.
Columns k=0-10 give: A238351, A240736, A240737, A240738, A240739, A240740, A240741, A240742, A240743, A240744, A240745.
The version for permutations is A008290.
The version with all zeros removed is A238350.
The version for reversed partitions is A238352.
The corresponding rank statistic is A352512, nonfixed A352513.
The version for nonfixed points is A352523, A352520 (k=1).
Below: comps = compositions, first = column k=0, stat = rank statistic.
- A352521 counts comps by strong nonexcedances, first A219282, stat A352514.
- A352522 counts comps by weak nonexcedances, first A238874, stat A352515.
- A352524 counts comps by strong excedances, first A008930, stat A352516.
- A352525 counts comps by weak excedances, A177510 (k=1), stat A352517.
Cf. A008292, A088218, A098825, A227682, A352487.

b:= proc(n, i) option remember; `if`(n=0, 1, expand(
       add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
seq(T(n), n=0..15);

b[n_, i_] := b[n, i] = If[n == 0, 1, Expand[Sum[b[n-j, i+1]*If[i == j, x, 1], {j, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 1]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)
pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pq[#]==k&]],{n,0,9},{k,0,n}] (* Gus Wiseman, Apr 03 2022 *)

A238350 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k parts p at position p (fixed points), n>=0, 0<=k<=A003056(n).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 1, 3, 4, 1, 6, 7, 3, 11, 16, 4, 1, 22, 29, 12, 1, 42, 60, 23, 3, 82, 120, 47, 7, 161, 238, 100, 12, 1, 316, 479, 198, 30, 1, 624, 956, 404, 61, 3, 1235, 1910, 818, 126, 7, 2449, 3817, 1652, 258, 16, 4864, 7633, 3319, 537, 30, 1, 9676, 15252, 6686, 1083, 70, 1, 19267, 30491, 13426, 2205

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 25 2014

			Triangle T(n,k) begins:
00 :   1;
01 :   0,   1;
02 :   1,   1;
03 :   2,   1,   1;
04 :   3,   4,   1;
05 :   6,   7,   3;
06 :  11,  16,   4,  1;
07 :  22,  29,  12,  1;
08 :  42,  60,  23,  3;
09 :  82, 120,  47,  7;
10 : 161, 238, 100, 12, 1;
11 : 316, 479, 198, 30, 1;
12 : 624, 956, 404, 61, 3;
     ...

M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.

Joerg Arndt and Alois P. Heinz, Rows n = 0..500, flattened
M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.

Columns k=0-10 give: A238351, A240736, A240737, A240738, A240739, A240740, A240741, A240742, A240743, A240744, A240745.
Row sums are A011782.
T(n*(n+3)/2,n) = A227682(n).
Same as A238349 without the trailing zeros.
Cf. A099036.

b:= proc(n, i) option remember; `if`(n=0, 1, expand(
       add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)):
seq(T(n), n=0..20);

b[n_, i_] := b[n, i] = If[n == 0, 1, Expand[Sum[b[n-j, i+1]*If[i == j, x, 1], {j, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

Sum_{k=0..A003056(n)} k * T(n,k) = A099036(n-1) for n>0.

A227681 G.f.: exp( Sum_{n>=1} x^n * (1+x)^n / (n*(1-x^n)) ).

Original entry on oeis.org

1, 1, 3, 6, 12, 23, 43, 79, 142, 252, 442, 766, 1316, 2244, 3799, 6393, 10704, 17841, 29618, 49000, 80823, 132964, 218242, 357501, 584608, 954553, 1556575, 2535425, 4125805, 6708143, 10898897, 17696749, 28719276, 46586050, 75538702, 122444483, 198420445, 321461918

Offset: 0

Paul D. Hanna, Jul 19 2013

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 12*x^4 + 23*x^5 + 43*x^6 + 79*x^7 +...
where
log(A(x)) = x*(1+x)/(1-x) + x^2*(1+x)^2/(2*(1-x^2)) + x^3*(1+x)^3/(3*(1-x^3)) + x^4*(1+x)^4/(4*(1-x^4)) + x^5*(1+x)^5/(5*(1-x^5)) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 10*x^3/3 + 17*x^4/4 + 26*x^5/5 + 38*x^6/6 + 57*x^7/7 + 81*x^8/8 + 118*x^9/9 + 180*x^10/10 +...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A160571, A227682, A306565, A306749.

nmax = 50; CoefficientList[Series[Product[1/(1 - x^k*(1 + x)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2021 *)

{a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*(1+x)^m/(1-x^m +x*O(x^n)) )), n)}
for(n=0, 50, print1(a(n), ", "))

{a(n)=polcoeff(exp(sum(m=1, n+1, x^m*sumdiv(m, d, (1+x +x*O(x^n))^d/d) )), n)}
for(n=0, 50, print1(a(n), ", "))

G.f.: exp( Sum_{n>=1} x^n * Sum_{d|n} (1+x)^d / d ).
G.f.: Product {n >= 1} 1/(1 - (1 + x)*x^n). - Peter Bala, Jan 20 2015
a(n) ~ phi^(n+1) / (sqrt(5)* A276987), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 16 2019

A307599 Expansion of Product_{k>=1} (1 - x^k/(1 - x)).

Original entry on oeis.org

1, -1, -2, -2, -1, 2, 6, 11, 15, 16, 11, -2, -26, -61, -105, -152, -192, -209, -183, -89, 98, 400, 830, 1385, 2035, 2715, 3314, 3668, 3556, 2703, 790, -2521, -7550, -14542, -23591, -34546, -46901, -59670, -71261, -79358, -80830, -71690, -47133, -1684, 70504, 175168, 317232

Offset: 0

Seiichi Manyama, Apr 17 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..500

Convolution inverse of A227682.

Cf. A126348, A307601, A307602.

m = 46; CoefficientList[Series[Product[1 - x^k/(1 - x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-x^k/(1-x)))

N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, x^k*sumdiv(k, d, 1/(d*(1-x)^d)))))

G.f.: exp( - Sum_{k>=1} x^k * Sum_{d|k} 1/(d*(1-x)^d)).

A253829 Triangular array with g.f. Product_{n >= 1} 1/(1 - x*z^n/(1 - z)).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 7, 4, 1, 0, 5, 13, 11, 5, 1, 0, 6, 22, 25, 16, 6, 1, 0, 7, 34, 50, 41, 22, 7, 1, 0, 8, 50, 91, 92, 63, 29, 8, 1, 0, 9, 70, 155, 187, 155, 92, 37, 9, 1, 0, 10, 95, 250, 353, 343, 247, 129, 46, 10, 1, 0, 11, 125, 386, 628, 701, 590, 376, 175, 56, 11, 1

Offset: 0

Peter Bala, Jan 19 2015

A refinement of A227682.
A colored composition of n is defined as a composition of n where each part p comes in one of p colors (denoted by an integer from 1 to p) and the color numbers are nondecreasing through the composition. The color numbers thus form a partition, called the color partition, of some integer.
For example, the composition 1 + 3 + 2 of 6 gives rise to three colored compositions of 6, namely, 1(c1) + 3(c1) + 2(c1), 1(c1) + 3(c1) + 2(c2) and 1(c1) + 3(c2) + 2(c2), where the color number of a part is shown after the part prefaced by the letter c.
T(n,k) equals the number of colored compositions of n into k parts.
See A253830 for the enumeration of colored compositions having parts with distinct colors.

			Triangle begins
n\k| 0  1   2   3   4   5   6  7
= = = = = = = = = = = = = = = = =
0  | 1
1  | 0  1
2  | 0  2   1
3  | 0  3   3   1
4  | 0  4   7   4   1
5  | 0  5  13  11   5   1
6  | 0  6  22  25  16   6  1
7  | 0  7  34  50  41  22  7  1
...
T(4,2) = 7: The compositions of 4 into two parts are 2 + 2, 1 + 3 and 3 + 1. Coloring the parts as described above produces seven colored compositions of 4 into two parts:
2(c1) + 2(c1), 2(c1) + 2(c2), 2(c2) + 2(c2),
1(c1) + 3(c1), 1(c1) + 3(c2), 1(c1) + 3(c3),
3(c1) + 1(c1).

P. Bala, Colored Compositions

Cf. A008284, A227682 (row sums), A253830.

G := 1/(product(1-x*z^j/(1-z), j = 1 .. 12)): Gser := simplify(series(G, z = 0, 14)): for n to 12 do P[n] := coeff(Gser, z^n) end do: for n to 12 do seq(coeff(P[n], x^j), j = 1 .. n) end do;

G.f.: G(x,z) := Product_{n >= 1} (1 - z)/(1 - z - x*z^n) = exp( Sum_{n >= 1} (x*z)^n/(n*(1 - z)^n*(1 - z^n)) ) =
1 + Sum_{n >= 1} (x*z/(1 - z))^n/( Product_{i = 1..n} 1 - z^i ) = 1 + x*z + (2*x + x^2)*z^2 + (3*x + 3*x^2 + x^3)*z^3 + ....
Note, G(x*(1 - z),z) is the generating function of A008284.
T(n,k) = Sum_{i = k..n} binomial(i-1,k-1)*A008284(n+k-i,k).
Recurrence equation: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-k,k) - T(n-k-1,k) with boundary conditions T(n,n) = 1, T(n,0) = 0 for n >= 1 and T(n,k) = 0 for n < k.
Row sums are A227682.

A307674 L.g.f.: log(Product_{k>=1} 1/(1 - x^k/(1 - x))) = Sum_{k>=1} a(k)*x^k/k.

Original entry on oeis.org

1, 5, 13, 29, 56, 107, 197, 365, 679, 1280, 2432, 4679, 9075, 17729, 34823, 68701, 135967, 269765, 536200, 1067284, 2126648, 4240978, 8462667, 16895039, 33742281, 67408931, 134697820, 269204657, 538104774, 1075723097, 2150667905, 4300088957, 8598178019

Offset: 1

Seiichi Manyama, Apr 21 2019

nonn

			L.g.f.: L(x) = x/1 + 5*x^2/2 + 13*x^3/3 + 29*x^4/4 + 56*x^5/5 + 107*x^6/6 + 197*x^7/7 + 365*x^8/8 + ... .
exp(L(x)) = 1 + x + 3*x^2 + 7*x^3 + 16*x^4 + 35*x^5 + 76*x^6 + 162*x^7 + 342*x^8 + ... + A227682(n)*x^n + ... .

Cf. A227682, A307599, A307675.

N=66; x='x+O('x^N); Vec(x*deriv(log(1/prod(k=1, N, 1-x^k/(1-x)))))

N=66; x='x+O('x^N); Vec(x*deriv(sum(k=1, N, x^k*sumdiv(k, d, 1/(d*(1-x)^d)))))

Product {k>=1} 1/(1 - x^k/(1 - x)) = exp(Sum_{k>=1} a(k)*x^k/k).

A307626 Expansion of Product_{k>=1} 1/(1 - x^k/(1 + x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 0, 8, -3, 16, -12, 36, -40, 88, -117, 220, -321, 560, -860, 1447, -2284, 3772, -6032, 9861, -15864, 25798, -41627, 67527, -109132, 176826, -285985, 463089, -749189, 1212722, -1962181, 3175635, -5138421, 8315361, -13455103, 21772865

Offset: 0

Seiichi Manyama, Apr 19 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..500

Convolution inverse A307601.

Cf. A227682.

m = 40; CoefficientList[Series[Product[1/(1 - x^k/(1 + x)), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, 1-x^k/(1+x)))

N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, 1/(d*(1+x)^d)))))

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} 1/(d*(1+x)^d)).

A322612 Expansion of e.g.f. Product_{k>=1} 1/(1 + log(1 - x)*x^k).

Original entry on oeis.org

1, 0, 2, 9, 68, 490, 5184, 53928, 696352, 9545184, 147901680, 2437886880, 44593856064, 861936989472, 17988878376000, 398199273907680, 9386173867046400, 233068382185213440, 6117261434418069504, 168414066137504272896, 4867992707164288773120, 147081824197157871866880, 4641822165217412602183680

Offset: 0

Ilya Gutkovskiy, Dec 20 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..427

Cf. A227682, A265953, A322613.

seq(coeff(series(factorial(n)*mul((1+log(1-x)*x^k)^(-1),k=1..n),x,n+1), x, n), n = 0 .. 22); # Muniru A Asiru, Dec 21 2018

nmax = 22; CoefficientList[Series[Product[1/(1 + Log[1 - x] x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Log[1/(1 - x)]^d/d, {d, Divisors[k]}] x^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} log(1/(1 - x))^d/d ) * x^k).

a(n) ~ c * n! / r^n, where r = 0.74075364335169502373416717320773551326074821766... is the root of the equation r*log(1-r) = -1 and c = 1 / (r*(r/(1-r) - log(1-r)) * Product_{k>=2} (1 + log(1-r)*r^k) ) = 16.634865259935976898139371781860039862... - Vaclav Kotesovec, Dec 20 2018

A336990 Expansion of Product_{k>=1} 1/(1 - x^k / (1 - k*x)).

Original entry on oeis.org

1, 1, 3, 8, 22, 62, 182, 560, 1822, 6316, 23467, 93762, 402989, 1858904, 9165303, 48065800, 266791060, 1560140592, 9573440644, 61431041813, 411006873603, 2859978776644, 20653331408062, 154494203986783, 1195107012223439, 9546189429869925, 78632580076861376, 667111706008969377

Offset: 0

Seiichi Manyama, Aug 10 2020

nonn

Cf. A227682, A336989, A336991.

m = 27; CoefficientList[Series[Product[1/(1 - x^k/(1 - k*x)), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Aug 10 2020 *)

N=40; x='x+O('x^N); Vec(1/prod(k=1, N, 1-x^k/(1-k*x)))

N=40; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, 1/(d*(1-k/d*x)^d)))))

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} 1 / (d * (1 - k/d * x)^d)).

cp's OEIS Frontend

A238349 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k parts p at position p (fixed points), n>=0, 0<=k<=n.

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A238350 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k parts p at position p (fixed points), n>=0, 0<=k<=A003056(n).

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A227681 G.f.: exp( Sum_{n>=1} x^n * (1+x)^n / (n*(1-x^n)) ).

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A307599 Expansion of Product_{k>=1} (1 - x^k/(1 - x)).

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A253829 Triangular array with g.f. Product_{n >= 1} 1/(1 - x*z^n/(1 - z)).

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A307674 L.g.f.: log(Product_{k>=1} 1/(1 - x^k/(1 - x))) = Sum_{k>=1} a(k)*x^k/k.

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A307626 Expansion of Product_{k>=1} 1/(1 - x^k/(1 + x)).

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