A268498 -id:A268498 - cp's OEIS frontend

A133121 Triangle T(n,k) read by rows = number of partitions of n such that number of parts minus number of distinct parts is equal to k, k = 0..n-1.

1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1, 4, 2, 3, 1, 0, 1, 5, 4, 2, 2, 1, 0, 1, 6, 6, 3, 3, 2, 1, 0, 1, 8, 7, 5, 4, 2, 2, 1, 0, 1, 10, 8, 10, 3, 5, 2, 2, 1, 0, 1, 12, 13, 8, 9, 4, 4, 2, 2, 1, 0, 1, 15, 15, 14, 10, 8, 5, 4, 2, 2, 1, 0, 1, 18, 21, 15, 16, 8, 9, 4, 4, 2, 2, 1, 0, 1, 22, 25, 23, 17, 17, 7, 10, 4, 4, 2, 2, 1, 0, 1

Offset: 1

Vladeta Jovovic, Sep 18 2007

			1
1,1
2,0,1
2,2,0,1
3,2,1,0,1
4,2,3,1,0,1
5,4,2,2,1,0,1
6,6,3,3,2,1,0,1
8,7,5,4,2,2,1,0,1
10,8,10,3,5,2,2,1,0,1
12,13,8,9,4,4,2,2,1,0,1
15,15,14,10,8,5,4,2,2,1,0,1
18,21,15,16,8,9,4,4,2,2,1,0,1
From _Gus Wiseman_, Jan 23 2019: (Start)
It is possible to augment the triangle to cover the n = 0 and k = n cases, giving:
   1
   1  0
   1  1  0
   2  0  1  0
   2  2  0  1  0
   3  2  1  0  1  0
   4  2  3  1  0  1  0
   5  4  2  2  1  0  1  0
   6  6  3  3  2  1  0  1  0
   8  7  5  4  2  2  1  0  1  0
  10  8 10  3  5  2  2  1  0  1  0
  12 13  8  9  4  4  2  2  1  0  1  0
  15 15 14 10  8  5  4  2  2  1  0  1  0
  18 21 15 16  8  9  4  4  2  2  1  0  1  0
  22 25 23 17 17  7 10  4  4  2  2  1  0  1  0
  27 30 32 21 19 16  8  9  4  4  2  2  1  0  1  0
Row seven {5, 4, 2, 2, 1, 0, 1, 0} counts the following integer partitions (empty columns not shown).
  (7)    (322)   (2221)  (22111)  (211111)  (1111111)
  (43)   (331)   (4111)  (31111)
  (52)   (511)
  (61)   (3211)
  (421)
(End)

Alois P. Heinz, Rows n = 1..141, flattened

Row sums are A000041. Row polynomials evaluated at -1 are A268498. Row polynomials evaluated at 2 are A006951.

Cf. A000009, A090858, A100471, A116608.

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
       add(x^`if`(j=0, 0, j-1)*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n$2)):
seq(T(n), n=1..16);  # Alois P. Heinz, Aug 21 2015

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[x^If[j == 0, 0, j-1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function [p, Table[ Coefficient[p, x, i], {i, 0, n - 1}]][b[n, n]]; Table[T[n], {n, 1, 16}] // Flatten (* Jean-François Alcover, Jan 23 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[#]-Length[Union[#]]==k&]],{n,0,15},{k,0,n}] (* augmented version, Gus Wiseman, Jan 23 2019 *)

partitm(n,m,nmin)={ local(resul,partj) ; if( n < 0 || m <0, return([;]) ; ) ; resul=matrix(0,m); if(m==0, return(resul); ) ; for(j=max(1,nmin),n\m, partj=partitm(n-j,m-1,j) ; for(r1=1,matsize(partj)[1], resul=concat(resul,concat([j],partj[r1,])) ; ) ; ) ; if(m==1 && n >= nmin, resul=concat(resul,[[n]]) ; ) ; return(resul) ; }
partit(n)={ local(resul,partm,filr) ; if( n < 0, return([;]) ; ) ; resul=matrix(0,n) ; for(m=1,n, partm=partitm(n,m,1) ; filr=vector(n-m) ; for(r1=1,matsize(partm)[1], resul=concat( resul,concat(partm[r1,],filr) ) ; ) ; ) ; return(resul) ; }
A133121row(n)={ local(p=partit(n),resul=vector(n),nprts,ndprts) ; for(r=1,matsize(p)[1], nprts=0 ; ndprts=0 ; for(c=1,n, if( p[r,c]==0, break, nprts++ ; if(c==1, ndprts++, if(p[r,c]!=p[r,c-1], ndprts++ ) ; ) ; ) ; ) ; k=nprts-ndprts; resul[k+1]++ ; ) ; return(resul) ; }
A133121()={ for(n=1,20, arow=A133121row(n) ; for(k=1,n, print1(arow[k],",") ; ) ; ) ; }
A133121() ; \\ R. J. Mathar, Sep 28 2007

tabl(nn) = my(pl = prod(n=1, nn, 1+x^n/(1-y*x^n)) + O(x^nn)); for (k=1, nn-1, print(Vecrev(polcoeff(pl,k,x)))); \\ Michel Marcus, Aug 23 2015

G.f.: Product_{n>=1} 1 + x^n/(1-y*x^n).

More terms from R. J. Mathar, Sep 28 2007

A268500 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 + x^k)).

Original entry on oeis.org

1, 0, 1, 2, 2, 6, 7, 14, 11, 42, 39, 70, 95, 142, 239, 378, 418, 624, 1106, 1200, 2250, 2836, 4166, 4902, 8021, 10410, 14961, 21268, 29477, 36714, 54172, 68358, 95071, 134946, 168035, 254190, 322335, 427338, 541054, 787264, 964969, 1340730, 1748094, 2311386

Offset: 0

Vaclav Kotesovec, Feb 06 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A022629, A267004, A267005, A268498, A268499.

nmax = 50; CoefficientList[Series[Product[(1+k*x^k)/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]

A104575 Alternating sum of diagonals in A060177.

Original entry on oeis.org

1, -1, -2, -1, -1, 3, 1, 7, 4, 4, 4, 2, -9, -7, -7, -28, -17, -25, -15, -24, -11, -8, 34, 19, 53, 46, 108, 110, 106, 113, 122, 108, 75, 103, -16, -87, -107, -169, -329, -257, -574, -501, -676, -609, -749, -588, -808, -548, -521, -315, -240, 369, 485, 865, 1099, 1738, 2129, 2686, 3088, 3460, 4103, 4011, 4480, 3983

Offset: 0

Vladeta Jovovic, Apr 21 2005

A090794(n) = (A000041(n)-a(n))/2. A092306(n) = (A000041(n)+a(n))/2.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A006951.

Cf. A000041, A008965, A081362, A092306, A268498.

CoefficientList[Series[Product[(1-2x^k)/(1-x^k),{k,70}],{x,0,70}],x] (* Harvey P. Dale, Jan 21 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-x^k/(1-x^k))) \\ Seiichi Manyama, Oct 05 2019

G.f.: Product_{i>0} (1 - 2*x^i)/(1 - x^i).

Euler transform of -A008965(n).

a(0)=1 prepended by Seiichi Manyama, Oct 05 2019

A327683 Expansion of Product_{k>0} (1+sqrt(1+4*x^k))/2.

Original entry on oeis.org

1, 1, 0, 4, -5, 17, -40, 144, -459, 1517, -5111, 17747, -62074, 219292, -782602, 2816664, -10205754, 37203230, -136360106, 502219652, -1857659296, 6897983144, -25704335380, 96090440940, -360265425619, 1354343161419, -5103948546609, 19278502980063, -72972099256954

Offset: 0

Seiichi Manyama, Sep 22 2019

sign

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

Convolution inverse of A327682.

Cf. A000009, A268498, A322204, A327684.

N:= 40:
P:= mul((1+sqrt(1+4*x^k))/2,k=1..N):
S:= series(P,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Sep 22 2019

nmax = 30; CoefficientList[Series[Product[(1+Sqrt[1+4*x^k])/2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 22 2019 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+sqrt(1+4*x^k))/2))

N=66; x='x+O('x^N); Vec(prod(i=1, N, 1-sum(j=1, N\i, (-1)^j*binomial(2*j-2, j-1)*x^(i*j)/j)))

a(n) ~ -(-1)^n * c * 2^(2*n - 1) / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=1} (1 + sqrt(1 + 4*(-1/4)^k))/2 = 0.52271977595412566689522667777276363119313248923... - Vaclav Kotesovec, May 06 2021

A268499 Expansion of Product_{k>=1} ((1 + 3*x^k) / (1 + x^k)).

Original entry on oeis.org

1, 2, 0, 8, -2, 8, 16, 8, 8, 10, 80, -8, 72, -24, 144, 128, 134, 40, 224, 120, 232, 688, 176, 696, 32, 1194, -96, 1840, 1144, 2248, 288, 2968, 800, 4160, 752, 5104, 6438, 4984, 5104, 5488, 10960, 4856, 14080, 3480, 24408, 15448, 26832, 7080, 42120, 11178

Offset: 0

Vaclav Kotesovec, Feb 06 2016

sign

In general, for m > 0, if g.f. = Product_{k>=1} ((1 + m*x^k) / (1 + x^k)) then a(n) ~ c^(1/4) * exp(sqrt(c*n)) / (2*sqrt((m+1)*Pi) * n^(3/4)), where c = Pi^2/3 + 2*log(m)^2 + 4*polylog(2, -1/m).

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A032308, A266821, A268498, A268500.

nmax = 100; CoefficientList[Series[Product[(1+3*x^k)/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c^(1/4) * exp(sqrt(c*n)) / (4*sqrt(Pi)*n^(3/4)), where c = Pi^2/3 + 2*log(3)^2 + 4*polylog(2, -1/3) = 4.467633549370382939364... .

A327684 Expansion of Product_{k>0} (1 + x^k/(1 + x^k/(1 + x^k))).

Original entry on oeis.org

1, 1, 0, 4, -4, 11, -13, 39, -73, 144, -256, 559, -1116, 2188, -4317, 8804, -17591, 34992, -69815, 140097, -280416, 560077, -1119327, 2240719, -4482527, 8961129, -17920037, 35847885, -71699202, 143384383, -286760131, 573549105, -1147115913, 2294173485, -4588309651, 9176739373

Offset: 0

Seiichi Manyama, Sep 22 2019

sign

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

Cf. A000009, A268498, A327683.

m = 35; CoefficientList[Series[Product[(1 + x^k/(1 + x^k/(1 + x^k))), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 06 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+3*x^k+x^(2*k))/(1+2*x^k)))

a(n) ~ -(-1)^n * c * 2^n, where c = 1/4 * Product_{k>=2} (1 + (-1/2)^k/(1 + (-1/2)^k/(1 + (-1/2)^k))) = 0.267077782295890034289082591596560646781284184591415208072736792505213482... - Vaclav Kotesovec, May 06 2021

cp's OEIS Frontend

A133121 Triangle T(n,k) read by rows = number of partitions of n such that number of parts minus number of distinct parts is equal to k, k = 0..n-1.

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A268500 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 + x^k)).

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A104575 Alternating sum of diagonals in A060177.

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A327683 Expansion of Product_{k>0} (1+sqrt(1+4*x^k))/2.

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A268499 Expansion of Product_{k>=1} ((1 + 3*x^k) / (1 + x^k)).

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A327684 Expansion of Product_{k>0} (1 + x^k/(1 + x^k/(1 + x^k))).

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