A217615 -id:A217615 - cp's OEIS frontend

A217661 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-x)^k.

1, 1, 2, 6, 16, 42, 114, 314, 870, 2426, 6804, 19168, 54198, 153730, 437232, 1246480, 3560838, 10190810, 29212432, 83860176, 241051796, 693709896, 1998535892, 5763312876, 16635018146, 48054500898, 138923916700, 401908892716, 1163493516356, 3370283517032

Offset: 0

Paul D. Hanna, Oct 10 2012

nonn

Radius of convergence of g.f. A(x) is |x| < 0.339332122592...
More generally, given
A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-t*x)^k then
A(x) = (1-t*x) / sqrt( (1-(t+1)*x)^2*(1+x^2) + (2*t-3)*x^2 - 2*t*(t-1)*x^3 ).

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 42*x^5 + 114*x^6 + 314*x^7 + ...
where the g.f. equals the series:
A(x) = 1 +
x*(1 + x/(1-x)) +
x^2*(1 + 2^2*x/(1-x) + x^2/(1-x)^2) +
x^3*(1 + 3^2*x/(1-x) + 3^2*x^2/(1-x)^2 + x^3/(1-x)^3) +
x^4*(1 + 4^2*x/(1-x) + 6^2*x^2/(1-x)^2 + 4^2*x^3/(1-x)^3 + x^4/(1-x)^4) +
x^5*(1 + 5^2*x/(1-x) + 10^2*x^2/(1-x)^2 + 10^2*x^3/(1-x)^3 + 5^2*x^4/(1-x)^4 + x^5/(1-x)^5) + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Colin Defant, James Propp, Quantifying Noninvertibility in Discrete Dynamical Systems, arXiv:2002.07144 [math.CO], 2020.

Cf. A217664, A217665, A217666, A217615.

CoefficientList[Series[(1-x)/Sqrt[1 - 4*x + 4*x^2 - 4*x^3 + 4*x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 17 2014 *)

a(n):=sum(sum(binomial(m,k)^2*binomial(n-m-1,n-m-k),k,0,n-m),m,0,n);

{a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k/(1-x +x*O(x^n))^k )), n)}
for(n=0,40,print1(a(n),", "))

G.f.: (1-x) / sqrt(1 - 4*x + 4*x^2 - 4*x^3 + 4*x^4).
a(n) ~ (1-r) / (2 * sqrt(Pi*n) * sqrt(1 - 3*r + 2*r^2 - r^3) * r^n), where r = 0.33933212259239... is the root of the equation 1-4*r+4*r^2-4*r^3+4*r^4 = 0. - Vaclav Kotesovec, Feb 17 2014
a(n) = Sum_{m=0..n} Sum_{k=0,n-m} C(m,k)^2*C(n-m-1,n-m-k). - Vladimir Kruchinin, Jan 16 2018

A217461 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(4-x)^(n-k).

Original entry on oeis.org

1, 4, 16, 72, 350, 1768, 9120, 47696, 251974, 1341560, 7186912, 38694000, 209187884, 1134838736, 6174666560, 33681995936, 184138474182, 1008642036184, 5534504908640, 30415064058160, 167378028670788, 922251663816368, 5087347689155264, 28091877168106592

Offset: 0

Paul D. Hanna, Oct 10 2012

nonn

Radius of convergence of g.f. is r = (3-sqrt(7))/2 = 0.1771243444...
More generally, given
A(x) = Sum_{n>=1} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k),
then A(x) = 1 / sqrt( (1 - t*x + 2*x^2)^2 - 4*x^2 )
and the radius of convergence r satisfies: (1-r)^2 = r*(t-r) for t>0.

			G.f.: A(x) = 1 + 4*x + 16*x^2 + 72*x^3 + 350*x^4 + 1768*x^5 +...
where the g.f. equals the series:
A(x) = 1 +
x*((4-x) + x) +
x^2*((4-x)^2 + 2^2*x*(4-x) + x^2) +
x^3*((4-x)^3 + 3^2*x*(4-x)^2 + 3^2*x^2*(4-x) + x^3) +
x^4*((4-x)^4 + 4^2*x*(4-x)^3 + 6^2*x^2*(4-x)^2 + 4^2*x^3*(4-x) + x^4) +
x^5*((4-x)^5 + 5^2*x*(4-x)^4 + 10^2*x^2*(4-x)^3 + 10^2*x^3*(4-x)^2 + 5^2*x^4*(4-x) + x^5) +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A217615, A217616, A217617.

CoefficientList[Series[1/Sqrt[(1-2*x+2*x^2)*(1-6*x+2*x^2)], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

{a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(4-x)^(m-k) + x*O(x^n))), n)}
for(n=0, 40, print1(a(n), ", "))

G.f.: A(x) = 1 / sqrt( (1 - 4*x + 2*x^2)^2 - 4*x^2 ).
G.f.: A(x) = 1 / sqrt( (1-2*x+2*x^2)*(1-6*x+2*x^2) ).
G.f. satisfies: A(x) = [1 + 2*x^2*Sum_{n>=0} A000108(n)*(-x*A(x))^(2*n)] / (1-4*x+2*x^2) where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan numbers.
Recurrence: n*a(n) = 4*(2*n-1)*a(n-1) - 16*(n-1)*a(n-2) + 8*(2*n-3)*a(n-3) - 4*(n-2)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ sqrt(147+56*sqrt(7))*(3+sqrt(7))^n/(14*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 24 2012

A217616 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(2-x)^(n-k).

Original entry on oeis.org

1, 2, 4, 12, 38, 116, 360, 1144, 3670, 11836, 38392, 125160, 409628, 1345000, 4428752, 14618608, 48356838, 160260332, 532009688, 1768729736, 5888250996, 19626282328, 65489004464, 218743627408, 731311554044, 2447018357208, 8194289368240, 27459924376592

Offset: 0

Paul D. Hanna, Oct 09 2012

nonn

Radius of convergence of g.f. is r = 1-sqrt(2)/2 = 0.2928932...
More generally, given
A(x) = Sum_{n>=1} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k),
then A(x) = 1 / sqrt( (1 - t*x + 2*x^2)^2 - 4*x^2 )
and the radius of convergence r satisfies: (1-r)^2 = r*(t-r) for t>0.

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 38*x^4 + 116*x^5 + 360*x^6 +...
where the g.f. equals the series:
A(x) = 1 +
x*((2-x) + x) +
x^2*((2-x)^2 + 2^2*x*(2-x) + x^2) +
x^3*((2-x)^3 + 3^2*x*(2-x)^2 + 3^2*x^2*(2-x) + x^3) +
x^4*((2-x)^4 + 4^2*x*(2-x)^3 + 6^2*x^2*(2-x)^2 + 4^2*x^3*(2-x) + x^4) +
x^5*((2-x)^5 + 5^2*x*(2-x)^4 + 10^2*x^2*(2-x)^3 + 10^2*x^3*(2-x)^2 + 5^2*x^4*(2-x) + x^5) +...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A217615, A217617, A217461, A216434.

CoefficientList[Series[1/Sqrt[(1-2*x+2*x^2)^2-4*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 17 2013 *)

{a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(2-x)^(m-k) + x*O(x^n))), n)}
for(n=0,40,print1(a(n),", "))

G.f.: A(x) = 1 / sqrt( (1 - 2*x + 2*x^2)^2 - 4*x^2 ).
G.f.: A(x) = 1 / sqrt( (1-4*x+2*x^2)*(1+2*x^2)  ).
G.f. satisfies: A(x) = [1 + 2*x^2*Sum_{n>=0} A000108(n)*(-x*A(x))^(2*n)] / (1-2*x+2*x^2) where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan
numbers.
Recurrence: n*a(n) = 2*(2*n-1)*a(n-1) - 4*(n-1)*a(n-2) + 4*(2*n-3)*a(n-3) - 4*(n-2)*a(n-4). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ sqrt(8+6*sqrt(2)) * (2+sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Sep 17 2013

A385572 Number of subsets of {1..n} with the same number of maximal runs (increasing by 1) as maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 2, 3, 4, 7, 12, 19, 34, 63, 112, 207, 394, 739, 1398, 2687, 5152, 9891, 19128, 37039, 71754, 139459, 271522, 528999, 1032308, 2017291, 3945186, 7723203, 15134440, 29679407, 58245068, 114389683, 224796210, 442021743, 869658304, 1711914351, 3371515306

Offset: 0

Gus Wiseman, Jul 04 2025

nonn

Also the number of subsets of {1..n} with the same number of adjacent elements increasing by 1 as adjacent elements increasing by more than 1.

			The set {2,3,5,6,8} has maximal runs ((2,3),(5,6),(8)) and maximal anti-runs ((2),(3,5),(6,8)) so is counted under a(8).
The a(0) = 1 through a(6) = 19 subsets:
  {}  {}   {}   {}   {}       {}       {}
      {1}  {1}  {1}  {1}      {1}      {1}
           {2}  {2}  {2}      {2}      {2}
                {3}  {3}      {3}      {3}
                     {4}      {4}      {4}
                     {1,2,4}  {5}      {5}
                     {1,3,4}  {1,2,4}  {6}
                              {1,2,5}  {1,2,4}
                              {1,3,4}  {1,2,5}
                              {1,4,5}  {1,2,6}
                              {2,3,5}  {1,3,4}
                              {2,4,5}  {1,4,5}
                                       {1,5,6}
                                       {2,3,5}
                                       {2,3,6}
                                       {2,4,5}
                                       {2,5,6}
                                       {3,4,6}
                                       {3,5,6}

Christian Sievers, Table of n, a(n) for n = 0..3328

The LHS is counted by A034839 (for partitions A384881, strict A116674), rank statistic A069010.
The case containing n + 1 is A217615.
The RHS is counted by A384893 or A210034 (for partitions A268193, strict A384905), rank statistic A384890.
Subsets of this type are ranked by A385575.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A000071, A000079, A001629, A010027, A053538, A384177, A384879, A384889.

a:= proc(n) option remember; `if`(n<5, [1, 2, 3, 4, 7][n+1], ((3*n-4)*a(n-1)-
      (3*n-5)*a(n-2)+(5*n-12)*a(n-3)-2*(4*n-11)*a(n-4)+4*(n-3)*a(n-5))/(n-1))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jul 06 2025

Table[Length[Select[Subsets[Range[n]],Length[Split[#,#2==#1+1&]]==Length[Split[#,#2!=#1+1&]]&]],{n,0,10}]

a(n)=polcoef([1,1,1]*[x,0,0;x,x^2,1;0,x,x]^n*[1,0,0]~,n) \\ Christian Sievers, Jul 06 2025

Let M be the matrix [1,0,0; 1,x,1/x; 0,1,1]. Then a(n) is the sum of the constant terms of the entries in the left column of M^n. - Christian Sievers, Jul 06 2025

a(21) and beyond from Christian Sievers, Jul 06 2025

A217421 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^3 * x^k*(1-x)^(n-k).

Original entry on oeis.org

1, 1, 1, 7, 19, 37, 151, 541, 1477, 4999, 18277, 58651, 195001, 694747, 2380429, 8091223, 28449667, 99709057, 346212319, 1217144833, 4300487251, 15137730889, 53484426379, 189831062953, 673630230931, 2393381462941, 8527594750507, 30416413412917, 108577772050459

Offset: 0

Paul D. Hanna, Oct 19 2012

nonn

			G.f.: A(x) = 1 + x + x^2 + 7*x^3 + 19*x^4 + 37*x^5 + 151*x^6 + 541*x^7 +...
where
A(x) = 1 +
x*((1-x) + x) +
x^2*((1-x)^2 + 2^3*x*(1-x) + x^2) +
x^3*((1-x)^3 + 3^3*x*(1-x)^2 + 3^3*x^2*(1-x) + x^3) +
x^4*((1-x)^4 + 4^3*x*(1-x)^3 + 6^3*x^2*(1-x)^2 + 4^3*x^3*(1-x) + x^4) +
x^5*((1-x)^5 + 5^3*x*(1-x)^4 + 10^3*x^2*(1-x)^3 + 10^3*x^3*(1-x)^2 + 5^3*x^4*(1-x) + x^5) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A217615.

CoefficientList[Series[HypergeometricPFQ[{1/3, 2/3},{1},27*x^3/(1-x)^2]/(1-x), {x, 0, 30}], x] (* Vaclav Kotesovec, Dec 08 2021 *)

{a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^3*x^k*(1-x)^(m-k) + x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))

G.f.: hypergeom([1/3, 2/3],[1],27*x^3/(1-x)^2)/(1-x). - Mark van Hoeij, Apr 11 2014

Recurrence: (n-4)*n^2*a(n) = (4*n^3 - 21*n^2 + 23*n - 9)*a(n-1) - (6*n^3 - 39*n^2 + 73*n - 45)*a(n-2) + (31*n^3 - 220*n^2 + 455*n - 279)*a(n-3) - (55*n^3 - 468*n^2 + 1278*n - 1125)*a(n-4) + 3*(n-3)*(3*n - 11)*(3*n - 10)*a(n-5). - Vaclav Kotesovec, Dec 08 2021

A217617 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(3-x)^(n-k).

Original entry on oeis.org

1, 3, 9, 33, 133, 549, 2295, 9711, 41505, 178749, 774387, 3370995, 14733043, 64608555, 284143257, 1252749777, 5535201733, 24503713893, 108659076723, 482566381299, 2146042722591, 9555487997247, 42594294578949, 190060286569677, 848858809506279, 3794468370955587

Offset: 0

Paul D. Hanna, Oct 09 2012

nonn

Radius of convergence of g.f. is r = (5-sqrt(17))/4 = 0.21922359...
More generally, given
A(x) = Sum_{n>=1} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k),
then A(x) = 1 / sqrt( (1 - t*x + 2*x^2)^2 - 4*x^2 )
and the radius of convergence r satisfies: (1-r)^2 = r*(t-r) for t>0.

			G.f.: A(x) = 1 + 3*x + 9*x^2 + 33*x^3 + 133*x^4 + 549*x^5 + 2295*x^6 +...
where the g.f. equals the series:
A(x) = 1 +
x*((3-x) + x) +
x^2*((3-x)^2 + 2^2*x*(3-x) + x^2) +
x^3*((3-x)^3 + 3^2*x*(3-x)^2 + 3^2*x^2*(3-x) + x^3) +
x^4*((3-x)^4 + 4^2*x*(3-x)^3 + 6^2*x^2*(3-x)^2 + 4^2*x^3*(3-x) + x^4) +
x^5*((3-x)^5 + 5^2*x*(3-x)^4 + 10^2*x^2*(3-x)^3 + 10^2*x^3*(3-x)^2 + 5^2*x^4*(3-x) + x^5) +...

Cf. A217615, A217616, A217461, A216454.

CoefficientList[Series[1/Sqrt[(1-3*x+2*x^2)^2-4*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 16 2013 *)

{a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(3-x)^(m-k) + x*O(x^n))), n)}
for(n=0,40,print1(a(n),", "))

G.f.: A(x) = 1 / sqrt( (1 - 3*x + 2*x^2)^2 - 4*x^2 ).
G.f.: A(x) = 1 / sqrt( (1-x+2*x^2)*(1-5*x+2*x^2) ).
G.f. satisfies: A(x) = [1 + 2*x^2*Sum_{n>=0} A000108(n)*(-x*A(x))^(2*n)] / (1-3*x+2*x^2) where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan
numbers.
Recurrence: n*a(n) = 3*(2*n-1)*a(n-1) - 9*(n-1)*a(n-2) + 6*(2*n-3)*a(n-3) - 4*(n-2)*a(n-4). - Vaclav Kotesovec, Sep 16 2013
a(n) ~ 2*((5+sqrt(17))/2)^n/sqrt((42*sqrt(17)-170)*Pi*n). - Vaclav Kotesovec, Sep 16 2013

A383503 Expansion of 1/sqrt( (1-x) * (1-x-4*x^3)^3 ).

Original entry on oeis.org

1, 2, 3, 10, 23, 42, 97, 218, 435, 918, 1977, 4062, 8393, 17590, 36303, 74614, 154211, 317334, 650505, 1335054, 2736453, 5595950, 11439475, 23370270, 47681965, 97217882, 198110199, 403383026, 820820215, 1669405626, 3393344257, 6893850650, 13999109715, 28414742790

Offset: 0

Seiichi Manyama, May 05 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A383254, A383499.

Cf. A217615, A377213.

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/Sqrt( (1-x) * (1-x-4*x^3)^3 ))); // Vincenzo Librandi, May 06 2025

Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[n-2*k+1,k+1],{k,0,Floor[n/3]}],{n,0,35}] (* Vincenzo Librandi, May 06 2025 *)

a(n) = sum(k=0, n\3, (2*k+1)*binomial(2*k, k)*binomial(n-2*k+1, k+1));

a(n) = Sum_{k=0..floor(n/3)} (2*k+1) * binomial(2*k,k) * binomial(n-2*k+1,k+1).

A385575 Numbers whose binary indices have the same number of adjacent parts differing by 1 as adjacent parts differing by more than 1.

Original entry on oeis.org

1, 2, 4, 8, 11, 13, 16, 19, 22, 25, 26, 32, 35, 38, 44, 49, 50, 52, 64, 67, 70, 76, 87, 88, 91, 93, 97, 98, 100, 104, 107, 109, 117, 128, 131, 134, 140, 151, 152, 155, 157, 167, 174, 176, 179, 182, 185, 186, 193, 194, 196, 200, 203, 205, 208, 211, 214, 217

Offset: 1

Gus Wiseman, Jul 04 2025

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their binary expansions and binary indices begin:
    1:       1 ~ {1}
    2:      10 ~ {2}
    4:     100 ~ {3}
    8:    1000 ~ {4}
   11:    1011 ~ {1,2,4}
   13:    1101 ~ {1,3,4}
   16:   10000 ~ {5}
   19:   10011 ~ {1,2,5}
   22:   10110 ~ {2,3,5}
   25:   11001 ~ {1,4,5}
   26:   11010 ~ {2,4,5}
   32:  100000 ~ {6}
   35:  100011 ~ {1,2,6}
   38:  100110 ~ {2,3,6}
   44:  101100 ~ {3,4,6}
   49:  110001 ~ {1,5,6}
   50:  110010 ~ {2,5,6}
   52:  110100 ~ {3,5,6}
   64: 1000000 ~ {7}
   67: 1000011 ~ {1,2,7}
   70: 1000110 ~ {2,3,7}
   76: 1001100 ~ {3,4,7}
   87: 1010111 ~ {1,2,3,5,7}
   88: 1011000 ~ {4,5,7}
   91: 1011011 ~ {1,2,4,5,7}
   93: 1011101 ~ {1,3,4,5,7}
   97: 1100001 ~ {1,6,7}
   98: 1100010 ~ {2,6,7}
  100: 1100100 ~ {3,6,7}

The LHS rank statistic is A069010, counted by A034839 (for partitions A384881, A116674).
The RHS rank statistic is A384890, counted by A384893 (for partitions A268193, A384905).
Subsets of this type are counted by A385572, with n A217615.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A000079, A010027, A053538, A210034, A384177, A384879, A384889.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],Length[Split[bpe[#],#2==#1+1&]]==Length[Split[bpe[#],#2!=#1+1&]]&]

is_ok(n)=hammingweight(n)==2*hammingweight(bitand(n,n>>1))+1 \\ Christian Sievers, Jul 18 2025

A383604 Expansion of 1/( (1-x)^2 * (1-x-9*x^3) )^(1/3).

Original entry on oeis.org

1, 1, 1, 4, 7, 10, 31, 70, 127, 328, 799, 1666, 4000, 9817, 22078, 52060, 126727, 296101, 699601, 1691350, 4024450, 9574393, 23081776, 55394488, 132650923, 319807159, 770872429, 1855190146, 4479086230, 10825202521, 26145137668, 63241928080, 153144714331

Offset: 0

Seiichi Manyama, May 01 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A383597, A383603.

Cf. A217615.

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/( (1-x)^2 * (1-x-9*x^3) )^(1/3))); // Vincenzo Librandi, May 06 2025

CoefficientList[Series[1/((1-x)^2*(1-x-9*x^3))^(1/3),{x,0,32}],x] (* Stefano Spezia, May 02 2025 *)
Table[Sum[(-9)^k*Binomial[-1/3,k]*Binomial[n-2*k,k],{k,0,Floor[n/3]}],{n,0,35}] (* Vincenzo Librandi, May 06 2025 *)

a(n) = sum(k=0, n\3, (-9)^k*binomial(-1/3, k)*binomial(n-2*k, k));

a(n) = Sum_{k=0..floor(n/3)} (-9)^k * binomial(-1/3,k) * binomial(n-2*k,k).

A385574 Number of integer partitions of n with the same number of adjacent equal parts as adjacent unequal parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 5, 6, 10, 11, 13, 17, 20, 30, 36, 44, 55, 70, 86, 98, 128, 156, 190, 235, 288, 351, 409, 499, 603, 722, 863, 1025, 1227, 1461, 1757, 2061, 2444, 2892, 3406, 3996, 4708, 5497, 6430, 7595, 8835, 10294, 12027, 13971, 16252, 18887, 21878

Offset: 0

Gus Wiseman, Jul 04 2025

nonn

These are also integer partitions of n with the same number of distinct parts as maximal anti-runs of parts.

			The partition (5,3,2,1,1,1,1) has 4 runs ((5),(3),(2),(1,1,1,1)) and 4 anti-runs ((5,3,2,1),(1),(1),(1)) so is counted under a(14).
The a(1) = 1 through a(10) = 10 reversed partitions (A = 10):
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)      (9)      (A)
                 (112)  (113)  (114)  (115)  (116)    (117)    (118)
                        (122)         (133)  (224)    (144)    (226)
                                      (223)  (233)    (225)    (244)
                                             (11123)  (11124)  (334)
                                                      (11223)  (11125)
                                                               (11134)
                                                               (11224)
                                                               (11233)
                                                               (12223)

Christian Sievers, Table of n, a(n) for n = 0..1000

The RHS is counted by A116608, rank statistic A297155.
The LHS is counted by A133121, rank statistic A046660.
For related inequalities see A212165, A212168, A361204.
For subsets instead of partitions see A217615, A385572, A385575.
These partitions are ranked by A385576.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A034839 counts subsets by number maximal runs, for partitions A384881, strict A116674.
A047993 counts partitions with max part = length (A106529).
A098859 counts Wilf partitions (complement A336866), compositions A242882.
A268193 counts partitions by maximal anti-runs, strict A384905, subsets A384893.
A355394 counts partitions with neighbors, complement A356236.
Cf. A000071, A003114, A008284, A010027, A047966, A210034, A325324, A325325, A356606, A384882, A384885.

Table[Length[Select[Reverse/@IntegerPartitions[n],Length[Union[#]]==Length[Split[#,#2!=#1&]]&]],{n,0,30}]

lista(n)=Vec(polcoef((prod(i=1,n,1+x^i/(t*(1-t*x^i))+O(x*x^n))-1)*t+1,0,t)) \\ Christian Sievers, Jul 18 2025

For a partition p, let s(p) be its sum, e(p) the number of equal adjacent pairs, and d(p) the number of distinct adjacent pairs. Then Sum_{p partition} x^s(p) * t^(e(p)-d(p)) = (Product_{i>=1} (1 + x^i/(t*(1-t*x^i))) - 1) * t + 1, so a(n) is the coefficient of x^n*t^0 of this expression. - Christian Sievers, Jul 18 2025

cp's OEIS Frontend

A217661 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k/(1-x)^k.

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A217461 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(4-x)^(n-k).

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A217616 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(2-x)^(n-k).

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A385572 Number of subsets of {1..n} with the same number of maximal runs (increasing by 1) as maximal anti-runs (increasing by more than 1).

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A217421 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^3 * x^k*(1-x)^(n-k).

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A217617 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(3-x)^(n-k).

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A383503 Expansion of 1/sqrt( (1-x) * (1-x-4*x^3)^3 ).

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