A143862 -id:A143862 - cp's OEIS frontend

A143773 Number of partitions of n such that every part is divisible by number of parts.

1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 5, 1, 4, 3, 6, 1, 8, 1, 7, 5, 6, 1, 14, 2, 7, 8, 11, 1, 17, 1, 14, 11, 9, 3, 29, 1, 10, 15, 23, 1, 28, 1, 23, 25, 12, 1, 51, 2, 20, 25, 32, 1, 44, 11, 39, 31, 15, 1, 94, 1, 16, 40, 52, 19, 64, 1, 57, 45, 44, 1, 126, 1, 19, 83, 74, 6, 90, 1, 124, 63, 21, 1, 186

Offset: 1

Vladeta Jovovic, Aug 31 2008

			The a(18) = 8 partitions are (18), (10 8), (12 6), (14 4), (16 2), (6 6 6), (9 6 3), (12 3 3). - _Gus Wiseman_, Jan 26 2018

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

Cf. A000005, A000041, A067538, A143862, A298422, A298423, A298426.

m = 100;
gf = Sum[x^(k^2)/Product[1-x^(k*i), {i, 1, k}], {k, 1, Sqrt[m]//Ceiling}];
CoefficientList[gf + O[x]^m, x] // Rest (* Jean-François Alcover, May 13 2019 *)

Vec(sum(k=1,20,x^(k^2)/prod(i=1,k,1-x^(k*i)+O(x^400)))) \\ Max Alekseyev, May 03 2009

G.f.: Sum(x^(k^2)/Product(1-x^(k*i), i=1..k), k=1..infinity).

For prime p, a(p) = 1 and a(p^2) = 2. For odd prime p, a(2*p) = (p + 1)/2. - Peter Bala, Mar 03 2025

More terms from Max Alekseyev, May 03 2009

A219282 Number of superdiagonal bargraphs with area n.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 35, 49, 68, 93, 126, 170, 229, 308, 413, 551, 731, 965, 1269, 1664, 2177, 2842, 3701, 4806, 6222, 8031, 10337, 13272, 17003, 21740, 27745, 35343, 44936, 57021, 72213, 91274, 115149, 145010, 182309, 228841, 286819, 358964, 448614, 559857, 697694

Offset: 0

Joerg Arndt, Dec 04 2012

nonn

Number of compositions n = p(1) + p(2) + ... + p(m) such that p(k) >= k (superdiagonal compositions), see example. - Joerg Arndt, Dec 19 2012

Number of (n-2)-bit binary strings in which the runs of ones are successively (1, 11, 111, 1111, ...), as in for example 00101100111011110011111000... To turn such a string into a composition, add 'X0 to the start of the empty string and the mark ' to the end, replace the runs 1, 11, 111,... with '01, '011, '0111, ... then consider the distances between the marks. - Andrew Woods, Jan 02 2015

			From _Joerg Arndt_, Dec 19 2012: (Start)
The a(9) = 18 compositions 9 = p(1) + p(2) + ... + p(m) such that p(k) >= k are
[ 1]  [ 1 2 6 ]
[ 2]  [ 1 3 5 ]
[ 3]  [ 1 4 4 ]
[ 4]  [ 1 5 3 ]
[ 5]  [ 1 8 ]
[ 6]  [ 2 2 5 ]
[ 7]  [ 2 3 4 ]
[ 8]  [ 2 4 3 ]
[ 9]  [ 2 7 ]
[10]  [ 3 2 4 ]
[11]  [ 3 3 3 ]
[12]  [ 3 6 ]
[13]  [ 4 2 3 ]
[14]  [ 4 5 ]
[15]  [ 5 4 ]
[16]  [ 6 3 ]
[17]  [ 7 2 ]
[18]  [ 9 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Vincenzo Librandi)
Margaret Archibald, Aubrey Blecher, Arnold Knopfmacher, and Stephan Wagner, Subdiagonal and superdiagonal compositions, Art Disc. Appl. Math. (2024). See p. 10.
Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562.

Cf. A063978 (compositions such that p(k) >= k-1 for k >= 2).
Cf. A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).
Cf. A008930 (subdiagonal compositions), A238875 (subdiagonal partitions), A010054 (subdiagonal partitions into distinct parts).
Cf. A098131 (compositions with smallest part >= number of parts; g.f. Sum_{k>=0} x^(k^2)/(1-x)^k).
Cf. A143862 (compositions with every part divisible by number of parts; g.f. Sum_{k>=0} x^(k^2) / (1 - x^k)^k).
Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Row sums of A305556.

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-j, i+1), j=i..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 28 2014

nmax = 50; CoefficientList[Series[Sum[x^(k*(k+1)/2) / (1-x)^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 05 2015 *)
b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, i+1], {j, i, n}]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

N=66; q='q+O('q^N);
gf=sum(n=0,N, q^(n*(n+1)/2) / (1-q)^n );
v=Vec(gf)

G.f.: Sum_{n>=0} q^(n*(n+1)/2) / (1-q)^n.

a(n) = Sum_{k=0..floor((sqrt(8*n+1)-3)/2)} C(n-1-C(k+1,2),k), for n >= 1.

A217668 G.f.: Sum_{n>=0} x^n*(1 + x^n)^n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 1, 5, 4, 6, 1, 14, 1, 8, 11, 13, 1, 25, 1, 22, 22, 12, 1, 61, 6, 14, 37, 50, 1, 77, 1, 73, 56, 18, 36, 175, 1, 20, 79, 211, 1, 135, 1, 188, 232, 24, 1, 421, 8, 236, 137, 313, 1, 307, 331, 422, 172, 30, 1, 1423, 1, 32, 295, 601, 716, 727, 1

Offset: 0

Paul D. Hanna, Oct 10 2012

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + x^3 + 3*x^4 + x^5 + 5*x^6 + x^7 + 5*x^8 +...
where we have the following series identity:
A(x) = 1 + x*(1+x) + x^2*(1+x^2)^2 + x^3*(1+x^3)^3 + x^4*(1+x^4)^4 + x^5*(1+x^5)^5  + x^6*(1+x^6)^6 + x^7*(1+x^7)^7 + x^8*(1+x^8)^8 + x^9*(1+x^9)^9 +...
A(x) = 1/(1-x) + x^2/(1-x^2)^2 + x^6/(1-x^3)^3 + x^12/(1-x^4)^4 + x^20/(1-x^5)^5 + x^30/(1-x^6)^6 + x^42/(1-x^7)^7 + x^56/(1-x^8)^8 +...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A217667, A217669, A217670, A143862.

terms = 100; Sum[x^n*(1 + x^n)^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)

{a(n,t=1)=polcoeff(sum(m=0,n,x^m*(t+x^m +x*O(x^n))^m),n)}
for(n=0,100,print1(a(n),", "))

{a(n,t=1)=local(A=1+x); A=sum(k=0, sqrtint(n+1), x^(k*(k+1))/(1 - t*x^(k+1) +x*O(x^n))^(k+1) ); polcoeff(A, n)}
for(n=0,100,print1(a(n),", ")) \\ Paul D. Hanna, Sep 13 2014

{a(n) = if(n==0,1, sumdiv(n,d, binomial(n/d,d-1)) )}
for(n=0,100,print1(a(n),", ")) \\ Paul D. Hanna, Apr 25 2018

G.f.: Sum_{n>=0} x^(n*(n+1)) / (1 - x^(n+1))^(n+1). - Paul D. Hanna, Sep 13 2014

a(n) = Sum_{d|n} binomial(n/d, d-1) for n>0 with a(0) = 1. - Paul D. Hanna, Apr 25 2018

A318636 Expansion of Sum_{n>=1} ( (1 + x^n)^n - 1 ).

Original entry on oeis.org

1, 2, 3, 5, 5, 9, 7, 14, 10, 20, 11, 31, 13, 35, 25, 45, 17, 74, 19, 70, 56, 77, 23, 161, 26, 104, 111, 154, 29, 261, 31, 222, 198, 170, 56, 536, 37, 209, 325, 496, 41, 623, 43, 605, 626, 299, 47, 1407, 50, 602, 731, 1092, 53, 1305, 517, 1443, 1026, 464, 59, 4002, 61, 527, 1429, 2381, 1352, 2596, 67, 3009, 1840, 2787, 71, 6719, 73, 740, 5378, 4655, 407, 5135, 79, 10118

Offset: 1

Paul D. Hanna, Sep 07 2018

nonn

			G.f.: A(x) = x + 2*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 9*x^6 + 7*x^7 + 14*x^8 + 10*x^9 + 20*x^10 + 11*x^11 + 31*x^12 + 13*x^13 + 35*x^14 + 25*x^15 + 45*x^16 + ...
such that
A(x) = x + (1 + x^2)^2 - 1 + (1 + x^3)^3 - 1 + (1 + x^4)^4 - 1 + (1 + x^5)^5 - 1 + (1 + x^6)^6 - 1 + (1 + x^7)^7-1 + ...
RELATED SERIES.
The g.f. A(x) equals following series at y = 1:
Sum_{n>=1} ((y + x^n)^n - y^n) = x + 2*y*x^2 + 3*y^2*x^3 + (4*y^3 + 1)*x^4 + 5*y^4*x^5 + (6*y^5 + 3*y)*x^6 + 7*y^6*x^7 + (8*y^7 + 6*y^2)*x^8 + (9*y^8 + 1)*x^9 + (10*y^9 + 10*y^3)*x^10 + 11*y^10*x^11 + (12*y^11 + 15*y^4 + 4*y)*x^12 + 13*y^12*x^13 + (14*y^13 + 21*y^5)*x^14 + (15*y^14 + 10*y^2)*x^15 + (16*y^15 + 28*y^6 + 1)*x^16 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1024 from Paul D. Hanna)

Cf. A143862, A318637, A318638.

{a(n) = polcoeff( sum(m=1,n, (x^m + 1 +x*O(x^n))^m - 1), n)}
for(n=1,100, print1(a(n),", "))

a(n) = Sum_{d|n} binomial(n/d,d). - Ridouane Oudra, May 02 2019

G.f.: Sum_{k >=1} x^(k^2)/(1-x^k)^(k+1). - Seiichi Manyama, Oct 30 2023

A261605 G.f.: Sum_{n=-oo..+oo} x^(n^2) / (1 - x^n)^n.

Original entry on oeis.org

1, 2, -1, 4, -1, 6, -6, 8, 2, 12, -15, 12, 8, 14, -28, 32, 22, 18, -55, 20, 34, 72, -66, 24, 44, 28, -91, 140, 62, 30, -205, 32, 209, 244, -153, 72, -98, 38, -190, 392, 443, 42, -518, 44, -1, 788, -276, 48, 506, 52, -451, 852, -196, 54, -1086, 728, 1636, 1180, -435, 60, -1691

Offset: 0

Paul D. Hanna, Aug 25 2015

			G.f.: A(x) = 1 + 2*x - x^2 + 4*x^3 - x^4 + 6*x^5 - 6*x^6 + 8*x^7 + 2*x^8 + 12*x^9 - 15*x^10 + 12*x^11 + 8*x^12 + 14*x^13 - 28*x^14 + 32*x^15 + 22*x^16 +...
where A(x) = 1 + N(x) + P(x) such that
N(x) = (x-1) + (x^2-1)^2 + (x^3-1)^3 + (x^4-1)^4 + (x^5-1)^5 + (x^6-1)^6 +...
P(x) = x/(1-x) + x^4/(1-x^2)^2 + x^9/(1-x^3)^3 + x^16/(1-x^4)^4 + x^25/(1-x^5)^5 +...
explicitly,
N(x) = x - 2*x^2 + 3*x^3 - 3*x^4 + 5*x^5 - 9*x^6 + 7*x^7 - 2*x^8 + 10*x^9 - 20*x^10 + 11*x^11 - x^12 + 13*x^13 - 35*x^14 + 25*x^15 + 13*x^16 +...
P(x) = x + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + x^7 + 4*x^8 + 2*x^9 + 5*x^10 + x^11 + 9*x^12 + x^13 + 7*x^14 + 7*x^15 + 9*x^16 +...+ A143862(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A143862, A260147, A261608.

with(numtheory):
seq(add( (-1)^(n/d+d)*binomial(d, n/d) + binomial(n/d-1, d-1), d in divisors(n)), n = 1..60); # Peter Bala, Mar 03 2025

{a(n) = polcoeff(sum(m=-n-2,n+2,x^(m^2)/(1-x^m +x*O(x^n))^m), n)}
for(n=0,60,print1(a(n),", "))

{a(n) = polcoeff(sum(m=-n-2,n+2,(x^m-1 +x*O(x^n))^m), n)}
for(n=0,60,print1(a(n),", "))

{a(n) = polcoeff(1/2 + sum(m=-n-2,n+2,x^(m^2)/(1+x^m +x*O(x^n))^(m+1)), n)}
for(n=0,60,print1(a(n),", "))

{a(n) = polcoeff(1/2 + sum(m=-n-2,n+2,x^m*(1+x^m +x*O(x^n))^(m-1)), n)}
for(n=0,60,print1(a(n),", "))

G.f.: Sum_{n=-oo..+oo} (x^n - 1)^n.
G.f.: 1/2 + Sum_{n=-oo..+oo} x^(n^2) / (1 + x^n)^(n+1).
G.f.: 1/2 + Sum_{n=-oo..+oo} x^n * (1 + x^n)^(n-1).
From Peter Bala, Mar 03 2025: (Start)
For n >= 1, a(n) = Sum_{d divides n} (-1)^(n/d+d)*binomial(d, n/d) + binomial(n/d-1, d-1).
For odd prime p, a(p) = p + 1, a(2*p) = - p*(p + 1)/2, a(p^2) = p^2 + 3. (End)

A303506 G.f.: Sum_{n>=1} (-1)^(n-1) * x^(n^2)/(1 - x^n)^n.

Original entry on oeis.org

1, 1, 1, 0, 1, -1, 1, -2, 2, -3, 1, -1, 1, -5, 7, -7, 1, 3, 1, -12, 16, -9, 1, 1, 2, -11, 29, -32, 1, 28, 1, -49, 46, -15, 16, -18, 1, -17, 67, -67, 1, 53, 1, -140, 162, -21, 1, -103, 2, 103, 121, -244, 1, 55, 211, -305, 154, -27, 1, -17, 1, -29, 219, -486, 496, -73, 1, -592, 232, 766, 1, -931, 1, -35, 1278, -852, 211, -529, 1, 322, 327, -39, 1, -1654, 1821, -41, 379, -1492, 1, 750, 925, -1584

Offset: 1

Paul D. Hanna, Apr 25 2018

sign

			G.f.: A(x) = x + x^2 + x^3 + x^5 - x^6 + x^7 - 2*x^8 + 2*x^9 - 3*x^10 + x^11 - x^12 + x^13 - 5*x^14 + 7*x^15 - 7*x^16 + x^17 + 3*x^18 + ...
such that
A(x) = x/(1-x) - x^4/(1-x^2)^2 + x^9/(1-x^3)^3 - x^16/(1-x^4)^4 + x^25/(1-x^5)^5 - x^36/(1-x^6)^6 + x^49/(1-x^7)^7 - x^64/(1-x^8)^8 +...

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A143862, A303340, A217668.

{a(n) = sumdiv(n,d, binomial(n/d-1, d-1) * (-1)^(d-1) )}
for(n=1,100, print1(a(n),", "))

a(n) = Sum_{d|n} binomial(n/d-1, d-1) * (-1)^(d-1) for n>=1.

A303560 Decimal expansion of constant A = Sum_{n>=1} 1 / (2^n - 1)^n.

Original entry on oeis.org

1, 1, 1, 4, 0, 4, 6, 3, 5, 1, 0, 3, 8, 0, 0, 3, 0, 4, 9, 6, 1, 4, 9, 9, 4, 2, 3, 6, 2, 0, 0, 1, 7, 7, 2, 4, 7, 5, 6, 5, 1, 4, 3, 1, 6, 5, 5, 5, 8, 3, 8, 9, 0, 2, 3, 0, 6, 5, 1, 1, 1, 4, 5, 4, 0, 1, 4, 8, 1, 8, 6, 8, 5, 5, 4, 9, 2, 1, 6, 4, 9, 6, 1, 0, 5, 8, 0, 3, 4, 5, 4, 6, 7, 1, 5, 3, 0, 9, 8, 5, 3, 4, 7, 8, 7, 7, 2, 7, 0, 3, 7, 2, 3, 0, 1, 8, 4, 2, 1, 7

Offset: 1

Paul D. Hanna, Apr 26 2018

			Constant A = 1.1140463510380030496149942362001772475651431655583890...
This constant equals the sum of the following infinite series.
(1) A = 1 + 1/3^2 + 1/7^3 + 1/15^4 + 1/31^5 + 1/63^6 + 1/127^7 + 1/255^8 + 1/511^9 + 1/1023^10 + 1/2047^11 + 1/4095^12 + 1/8191^13 + 1/16383^14 + ...
Also,
(2) A = 1/2 + 5/2^4 + 9^2/2^9 + 17^3/2^16 + 33^4/2^25 + 65^5/2^36 + 129^6/2^49 + 257^7/2^64 + 513^8/2^81 + 1025^9/2^100 + 2049^10/2^121 + 4097^11/2^144 + ...
Expressed in terms of powers of 1/2, we have
(3) A = 1/2 + 1/2^2 + 1/2^3 + 2/2^4 + 1/2^5 + 3/2^6 + 1/2^7 + 4/2^8 + 2/2^9 + 5/2^10 + 1/2^11 + 9/2^12 + 1/2^13 + 7/2^14 + 7/2^15 + 9/2^16 + 1/2^17 + 19/2^18 + 1/2^19 + 14/2^20 + 16/2^21 + 11/2^22 + ... + A143862(n)/2^n + ...
DECIMAL EXPANSION TO 1000 DIGITS:
A = 1.11404635103800304961499423620017724756514316555838\
90230651114540148186855492164961058034546715309853\
47877270372301842177485420929460338909110702521744\
10383049196253371844115566456211414378684927895066\
60974873819605352670009454376709247947228660654797\
93238935770616752469239881642090329202510251771440\
45431299113929370687739805515426044704234082381940\
40977172853717815297745712948744536180513363052564\
03854647492812806063479313722184475876462500578835\
52045304926771113275210795841642087115096877536105\
78958100347787242164699010158545554930990338272655\
43040184293715344496344121360156193744995971261933\
73889789233802551892919983961109429243561889220300\
14247648527081291849257683339708248465152426049641\
37604659963590944969427064766226893075229683791387\
71510378240823470806036647280652258639308495696013\
02313861338203801779994141266165775176960964298159\
94404826055321122142831544652111697011832111972164\
78072293762844331943953407090067036379926709332730\
54952123380671191947076262400344800454366565708630...

Cf. A303561 (binary), A302765, A143862.

This constant may be defined by the following expressions.
(1) A = Sum_{n>=1} 1 / (2^n - 1)^n.
(2) A = Sum_{n>=1} (2^n + 1)^(n-1) / 2^(n^2).
(3) A = Sum_{n>=1} A143862(n)/2^n where A143862(n) = Sum_{d|n} binomial(n/d-1, d-1) for n>=1.

A338271 a(n) is the number of compositions of n, b_1 + ... + b_t = n such that sqrt(b_1 + sqrt(b_2 + ... + sqrt(b_t)...)) is an integer.

Original entry on oeis.org

1, 0, 0, 2, 0, 2, 0, 2, 2, 4, 2, 6, 2, 8, 4, 14, 6, 20, 8, 28, 14, 44, 20, 66, 30, 96, 46, 146, 70, 220, 102, 326, 154, 490, 232, 740, 346, 1102, 520, 1652, 782, 2484, 1166, 3716, 1750, 5568, 2628, 8358, 3936, 12518, 5900, 18760, 8848, 28138, 13256, 42170

Offset: 1

Peter Kagey, Oct 19 2020

nonn

a(n) <= Sum_{k=1..floor(sqrt(n)/2)} A338286(floor((n-4*k^2)/2)) when n is even.

a(n) <= Sum_{k=1..floor((sqrt(n) - 1)/2)} A338286(floor((n-4*k^2-4*k-1)/2)) when n is odd and greater than 1.

			(Let s(k) = sqrt(k) for brevity.)
For n = 14, the a(14) = 8 valid compositions are:
14 = 2+2+2+2+2+3+1 and 2 = s(2+s(2+s(2+s(2+s(2+s(3+s(1)))))))
14 = 1+7+2+3+1     and 2 = s(1+s(7+s(2+s(3+s(1)))))
14 = 2+1+7+3+1     and 2 = s(2+s(1+s(7+s(3+s(1)))))
14 = 2+2+1+8+1     and 2 = s(2+s(2+s(1+s(8+s(1)))))
14 = 2+2+2+2+2+4   and 2 = s(2+s(2+s(2+s(2+s(2+s(4))))))
14 = 1+7+2+4       and 2 = s(1+s(7+s(2+s(4))))
14 = 2+1+7+4       and 2 = s(2+s(1+s(7+s(4))))
14 = 2+2+1+9       and 2 = s(2+s(2+s(1+s(9))))

Peter Kagey, Table of n, a(n) for n = 1..500
Code Golf Stack Exchange, The square root of the square root of the square root of the...

Cf. A000196, A338268.

Cf. A098504, A101391, A143862, A238351, A238686, A325676. A335942.

a(n) = Sum_{i=k..A000196(n)} A338268(n,k).

A376018 a(n) = Sum_{d|n} d^d * binomial(n/d-1,d-1).

Original entry on oeis.org

1, 1, 1, 5, 1, 9, 1, 13, 28, 17, 1, 102, 1, 25, 163, 285, 1, 303, 1, 1061, 406, 41, 1, 3172, 3126, 49, 757, 5173, 1, 16654, 1, 9021, 1216, 65, 46876, 62546, 1, 73, 1783, 130956, 1, 282123, 1, 30805, 221208, 89, 1, 1024944, 823544, 393847, 3241, 56421, 1, 2616513

Offset: 1

Seiichi Manyama, Sep 06 2024

nonn

Cf. A143862, A376014.

a(n) = sumdiv(n, d, d^d*binomial(n/d-1, d-1));

my(N=60, x='x+O('x^N)); Vec(sum(k=1, N, (k*x^k/(1-x^k))^k))

from math import comb
from itertools import takewhile
from sympy import divisors
def A376018(n): return sum(d**d*comb(n//d-1,d-1) for d in takewhile(lambda d:d**2<=n,divisors(n))) # Chai Wah Wu, Sep 06 2024

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

If p is prime, a(p) = 1.

A299042 G.f.: Sum_{n>=0} x^(n^2) * C(x^n)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 1, 2, 6, 14, 44, 132, 434, 1431, 4876, 16796, 58831, 208012, 743032, 2674449, 9695275, 35357670, 129646248, 477638700, 1767268056, 6564120510, 24466283816, 91482563640, 343059672747, 1289904147325, 4861946609464, 18367353073153, 69533551658952, 263747951750360, 1002242219329245, 3814986502092304, 14544636048921919, 55534064877060132, 212336130447600780

Offset: 0

Paul D. Hanna, Feb 16 2018

nonn

Compare to: Sum{n>=0} Series_Reversion( x/(1 + x^n)^(1/n) )^(n^2) = Sum_{n>=0} x^(n^2)/(1 - x^n)^n, the g.f. of A143862.

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 14*x^5 + 44*x^6 + 132*x^7 + 434*x^8 + 1431*x^9 + 4876*x^10 + 16796*x^11 + 58831*x^12 + ...
such that
A(x) = 1 + (1 - sqrt(1 - 4*x))/2 + (1 - sqrt(1 - 4*x^2))^2/2^2 + (1 - sqrt(1 - 4*x^3))^3/2^3 + (1 - sqrt(1 - 4*x^4))^4/2^4 + (1 - sqrt(1 - 4*x^5))^5/2^5 + (1 - sqrt(1 - 4*x^6))^6/2^6 + ...
The related series x^(n^2) * C(x^n)^n = (1 - sqrt(1 - 4*x^n))^n/2^n begin:
n=1: x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + ...;
n=2: x^4 + 2*x^6 + 5*x^8 + 14*x^10 + 42*x^12 + ...;
n=3: x^9 + 3*x^12 + 9*x^15 + 28*x^18 + 90*x^21 + ...;
n=4: x^16 + 4*x^20 + 14*x^24 + 48*x^28 + 165*x^32 + ...;
n=5: x^25 + 5*x^30 + 20*x^35 + 75*x^40 + 275*x^45 + ...;
n=6: x^36 + 6*x^42 + 27*x^48 + 110*x^54 + 429*x^60 + ...;
...
SPECIFIC VALUES.
A(1/4) = Sum_{n>=0} (2^(n-1) - sqrt(4^(n-1) - 1))^n / 2^(n^2) = 1.504491300666... = 1 + 1/2 + (2 - sqrt(3))^2/2^4 + (4 - sqrt(15))^3/2^9 + (8 - sqrt(63))^4/2^16 + (16 - sqrt(255))^5/2^25 + (32 - sqrt(1023))^6/2^36 + (64 - sqrt(4095))^7/2^49 + ...
A(-1/4) = Sum_{n>=0} (2^(n-1) - sqrt(4^(n-1) + 1))^n / 2^(n^2) = 0.79637258079... = 1 + (1 - sqrt(2))/2 + (2 - sqrt(5))^2/2^4 + (4 - sqrt(17))^3/2^9 + (8 - sqrt(65))^4/2^16 + (16 - sqrt(257))^5/2^25 + (32 - sqrt(1025))^6/2^36 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A000108.

{a(n) = my(A); A = sum(m=0,sqrtint(n+1), (1 - sqrt(1 - 4*x^m +x*O(x^n) ))^m / 2^m); polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))

G.f.: Sum{n>=0} (1 - sqrt(1 - 4*x^n))^n / 2^n.

G.f.: Sum{n>=0} Series_Reversion( x*(1 - x^n)^(1/n) )^(n^2).

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