A276428 -id:A276428 - cp's OEIS frontend

A276427 Irregular triangle read by rows: T(n,k) = number of partitions of n having k distinct parts i of multiplicity i; 0 <= k <= A328806(n)-1 = largest index of a nonzero value; n >= 0.

1, 0, 1, 2, 2, 1, 3, 2, 5, 1, 1, 8, 3, 9, 6, 16, 5, 1, 19, 10, 1, 29, 11, 2, 36, 18, 2, 53, 21, 3, 65, 32, 4, 92, 38, 4, 1, 115, 54, 7, 154, 67, 10, 195, 88, 14, 257, 112, 15, 1, 318, 152, 19, 1, 419, 178, 29, 1, 516, 243, 31, 2, 663, 293, 44, 2, 821, 376, 56, 2, 1039, 465, 67, 4, 1277, 589, 89, 3, 1606, 715, 108, 7

Offset: 0

Emeric Deutsch, Sep 19 2016

The sum of entries in row n is A000041(n): the partition numbers. [This allows us to know the row length, i.e., when the largest value of k is reached for which T(n,k) is nonzero. The row lengths are now listed as A328806. - M. F. Hasler, Oct 28 2019]

			Triangle starts:
1;       (n=0: partition [] has k=0 parts i of multiplicity i: T(0,0) = 1.)
0, 1;    (n=1: partition [1] has k=1 part i of multiplicity i: T(1,1) = 1.)
2;       (n=2: partitions [1,1] and [2] have k=0 parts i occurring i times.)
2, 1;    (n=3: [1,1,1] and [3] have 0, [1,2] has 1 part i occurring i times)
3, 2;    (n=4: [4], [1,1,2] and [1,1,1,1] for k=0; [1,3] & [2,2] for k=1.)
5, 1, 1; (n=5: [1,4] has i=1, [1,2,2] has i=1 and i=2 occurring i times.)
(...)
The partition [1,2,3,3,3,4] has 2 parts i of multiplicity i: i=1 and i=3.
T(14,3) = 1, since [1,2,2,3,3,3] is the only partition of 14 having k=3 parts i with multiplicity i, namely i = 1, 2 and 3.
T(14,2) = 4, counting [1,2,2,3,6], [1,2,2,4,5], [1,2,2,9] (with i=1 and i=2), and [1,3,3,3,4] (with i=1 and i=3).

Alois P. Heinz, Rows n = 0..1000, flattened

Cf. A000041 (row sums), A276428, A276429, A328806 (row lengths).

G := mul((t-1)*x^(i^2)+1/(1-x^i), i = 1 .. 100): Gser := simplify(series(G, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(Gser, x, n)) end do: for n from 0 to 30 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(i=j, x, 1)*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Sep 19 2016

b[n_, i_] := b[n, i] = Expand[If[n==0, 1, If[i<1, 0, Sum[If[i==j, x, 1] * b[n-i*j, i-1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[ p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Oct 20 2016, after Alois P. Heinz *)

apply( A276427_row(n, r=List(0))={forpart(p=n, my(s, c=1); for(i=1, #p, p[i]==if(i<#p, p[i+1]) && c++ && next; c==p[i] && s++; c=1); while(#r<=s, listput(r,0)); r[s+1]++);Vec(r)}, [0..20]) \\ M. F. Hasler, Oct 27 2019

G.f.:  G(t,x) = Product_{i>=1} ((t-1)*x^{i^2} + 1/(1-x^i)).
T(n,0) = A276429(n).
Sum(k*T(n,k), k>=0) = A276428(n).

Name edited by M. F. Hasler, Oct 27 2019

A276434 Sum over all partitions of n of the number of distinct parts i of multiplicity i+1.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 3, 5, 6, 10, 12, 19, 23, 34, 41, 58, 72, 98, 121, 162, 200, 262, 323, 415, 511, 650, 796, 1000, 1222, 1522, 1851, 2287, 2771, 3399, 4103, 5000, 6015, 7289, 8735, 10530, 12579, 15094, 17968, 21468, 25477, 30319, 35873, 42531, 50177, 59291

Offset: 0

Emeric Deutsch, Sep 30 2016

nonn

			a(6) = 3 because in the partitions [1,1,1,1,1,1], [1,1,1,1,2], [1',1,2,2], [2',2,2], [1,1,1,3], [1,2,3], [3,3], [1',1,4], [2,4], [1,5], [6] of 6 only the marked parts satisfy the requirement.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A276427, A276428, A276429, A276433, A277099, A277100, A277101, A277102.

g := (sum(x^(i*(i+1))*(1-x^i), i = 1 .. 200))/(product(1-x^i, i = 1 .. 200)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
     `if`(i<1, 0, add((p-> p+`if`(i+1<>j, 0,
      [0, p[1]]))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 30 2016

max = 60; s = Sum[x^(i*(i+1))*(1-x^i), {i, 1, max}]/QPochhammer[x] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 08 2016 *)

a(n) = Sum(k*A276433(n,k), k>=0).

G.f.: g(x) = Sum_(i>=1) (x^(i(i+1))(1-x^i))/Product_(i>=1) (1-x^i).

A277101 Sum over all partitions of n of the number of distinct parts i of multiplicity i - 1.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 4, 5, 8, 10, 15, 20, 29, 37, 52, 67, 89, 115, 152, 192, 251, 316, 405, 508, 644, 799, 1006, 1243, 1546, 1901, 2351, 2871, 3527, 4289, 5232, 6336, 7688, 9264, 11189, 13430, 16137, 19299, 23097, 27514, 32799, 38944, 46246, 54738, 64782, 76430, 90171

Offset: 0

Emeric Deutsch, Oct 10 2016

nonn

			a(6) = 4 because in the partitions [1,1,1,1,1,1], [1,1,1,1,2'], [1,1,2,2], [2,2,2], [1,1,1,3], [1,2',3], [3',3], [1,1,4], [2',4], [1,5], [6] of 6 only the marked parts satisfy the requirement.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A276427, A276428, A276429, A276433, A276434, A277099, A277100, A277102.

g := (sum(x^(i*(i+1))*(1-x^(i+1)), i = 1 .. 200))/(product(1-x^i, i = 1 .. 200)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
     `if`(i<1, 0, add((p-> p+`if`(i-1<>j, 0,
      [0, p[1]]))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 10 2016

max = 60; s = Sum[x^(i*(i+1))*(1-x^(1+i)), {i, 1, max}]/QPochhammer[x] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 08 2016 *)

a(n) = Sum(k*A277100(n,k), k>=0).

G.f.: g(x) = Sum_(i>=1)(x^(i(i+1))(1-x^(i+1)))/Product_(i>=1)(1-x^i).

A276433 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i of multiplicity i+1 (n>=0).

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 1, 6, 1, 8, 3, 12, 3, 18, 3, 1, 24, 6, 32, 10, 45, 10, 1, 59, 17, 1, 79, 21, 1, 104, 28, 3, 137, 37, 2, 177, 50, 4, 229, 64, 4, 295, 82, 8, 377, 105, 8, 477, 139, 10, 1, 605, 174, 13, 761, 220, 21, 956, 275, 24, 1193, 350, 31, 1

Offset: 0

Emeric Deutsch, Sep 30 2016

Sum of entries in row n is A000041(n) (the partition numbers).
T(n,0) = A277099(n).
Sum(k*T(n,k), k>=0) = A276434(n).

			The partition [1,1,3,3,3,3,4] has 2 parts i of multiplicity i+1: 1 and 3.
T(5,1) = 1, counting [1,1,3].
T(6,1) = 3, counting [1,1,4], [1,1,2,2], and [2,2,2].
T(8,2) = 1, counting [1,1,2,2,2].
Triangle starts:
1;
1;
1,1;
3;
4,1;
6,1;
8,3.

Alois P. Heinz, Rows n = 0..1000, flattened

Cf. A000041, A276427, A276428, A276429, A276434, A277099, A277100, A277101, A277102.

G := mul((t-1)*x^(i*(i+1))+1/(1-x^i), i = 1 .. 100): Gser := simplify(series(G, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(Gser, x, n)) end do: for n from 0 to 30 do seq(coeff(P[n],t,k),k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(i+1=j, x, 1)*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Sep 30 2016

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[If[i + 1 == j, x, 1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Nov 28 2016 after Alois P. Heinz *)

G.f.: G(t,x) = Product_{i>=1} ((t-1)*x^(i(i+1)) + 1/(1-x^i)).

A277099 Number of partitions of n containing no part i of multiplicity i+1.

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 8, 12, 18, 24, 32, 45, 59, 79, 104, 137, 177, 229, 295, 377, 477, 605, 761, 956, 1193, 1484, 1840, 2276, 2800, 3441, 4210, 5141, 6261, 7603, 9206, 11132, 13419, 16144, 19380, 23223, 27763, 33134, 39467, 46931, 55703, 66008, 78085, 92239, 108776, 128091, 150617

Offset: 0

Emeric Deutsch, Sep 30 2016

nonn

			a(4) = 4 because we have [1,1,1,1], [1,3], [2,2], and [4]; the partition [1,1,2] does not qualify.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A276427, A276428, A276429, A276433, A276434, A277100, A277101, A277102.

g:= product(1/(1-x^i)-x^(i*(i+1)), i = 1 .. 100): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(i+1=j, 0, b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 30 2016

nmax = 50; CoefficientList[Series[Product[(1/(1-x^k) - x^(k*(k+1))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 30 2016 *)

a(n) = A276433(n,0).

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

A277100 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i (i>=2) of multiplicity i-1 (n>=0).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 5, 2, 7, 4, 10, 5, 15, 6, 1, 21, 8, 1, 28, 13, 1, 37, 18, 1, 50, 25, 2, 67, 31, 3, 88, 42, 5, 115, 55, 6, 150, 73, 8, 193, 93, 11, 248, 122, 15, 317, 154, 19, 402, 200, 24, 1, 508, 253, 30, 1, 640, 320, 41, 1, 802, 399, 53, 1, 1002, 503, 69, 1

Offset: 0

Emeric Deutsch, Oct 10 2016

Sum of entries in row n is A000041(n) (the partition numbers).
T(n,0) = A277102(n).
Sum(k*T(n,k), k>=0) = A277101(n).

			The partition [1,1,2,3,3,3,3,4,4,4] has 2 parts i of multiplicity i-1: 2 and 4.
T(5,1) = 2, counting [1,1,1,2] and [2,3].
T(8,2) = 1, counting [2,3,3].
Triangle starts:
1;
1;
1, 1;
2, 1;
4, 1;
5, 2;
7, 4;
...

Alois P. Heinz, Rows n = 0..1000, flattened

Cf. A000041, A116608, A276427, A276428, A276429, A276433, A276434, A277099, A277101, A277102.

g := mul((t-1)*x^(i*(i+1))+1/(1-x^(i+1)), i = 1 .. 100)/(1-x): gser := simplify(series(g, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 30 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(i-1=j, x, 1)*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Oct 10 2016

b[n_, i_] := b[n, i] = Expand[If[n==0, 1, If[i<1, 0, Sum[If[i-1 == j, x, 1]*b[n-i*j, i-1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 08 2016 after Alois P. Heinz *)

G.f.: G(t,x) = Product_{i>=1} ((t-1)*x^(i(i+1)) + 1/(1-x^(i+1))).

A277102 Number of partitions of n containing no part i of multiplicity i-1.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 7, 10, 15, 21, 28, 37, 50, 67, 88, 115, 150, 193, 248, 317, 402, 508, 640, 802, 1002, 1248, 1545, 1908, 2351, 2887, 3532, 4313, 5251, 6377, 7724, 9334, 11254, 13541, 16253, 19473, 23286, 27791, 33100, 39362, 46723, 55370, 65504, 77377, 91257, 107477, 126380

Offset: 0

Emeric Deutsch, Oct 10 2016

nonn

			a(4) = 4 because we have [1,1,1,1], [1,3], [2,2], and [4]; the partition [1,1,2] does not qualify.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Alois P. Heinz)

Cf. A276427, A276428, A276429, A276433, A276434, A277099, A277100, A277101.

g := (product(1/(1-x^(i+1))-x^(i*(i+1)), i = 1 .. 100))/(1-x): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(i-1=j, 0, b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 10 2016

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[If[i-1 == j, 0, b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 11 2016 after Alois P. Heinz *)

a(n) = A277100(n,0).

G.f.: g(x) = Product_{i>=1}(1/(1-x^(i+1)) - x^(i(i+1))).

A328806 Row lengths of A276427: largest k such that a partition of n has k-1 distinct parts i of multiplicity i.

Original entry on oeis.org

1, 2, 1, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

M. F. Hasler, Oct 27 2019

nonn

Columns of A276427 are numbered starting with 0, so the row length is one more than the index of the last column.

			For n = 0, the empty partition [] has 0 parts i with multiplicity i, so a(0) = 1.
For n = 1, the partition [1] has one part i with multiplicity i, whence a(1) = 2.
For n = 2, both partitions [1,1] and [2] have 0 parts i with multiplicity i, so a(2) = 1.
For n = 3, the partition [1,2] has one part i with multiplicity i, hence a(3) = 2.
For n = 4, the partitions [1,3] and [2,2] have one part i with multiplicity i, so a(4) = 2.
For n = 5, the partition [1,2,2] has 2 parts i with multiplicity i, hence a(5) = 3.
The smallest partition with k-1 = 3 parts i with multiplicity i is [1,2,2,3,3,3], for n = 14, whence a(14) = 4.

Cf. A276427, A276428, A276429, A000041.

PARI
```
a(n)=#A276427_row(n)
```

More terms from Alois P. Heinz, Oct 28 2019

A328891 Irregular table T(n,k) = #{m > 0: m occurs m times in the k-th partition of n, using A&S order (A036036)}, 1 <= k <= A000041(n), n >= 0.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1

Offset: 0

M. F. Hasler, Oct 29 2019

In the n-th row, the partitions of n are considered in the "Abramowitz and Stegun" or graded (reflected or not) colexicographic ordering, as in A036036 or A036037. For each partition this counts the numbers m > 0 such that there are exactly m parts equal to m in the partition.

Row lengths are A000041(n) = number of partitions of n, the partition numbers.

			The table reads:
  n \ T(n,k), ...
  0 : 0;   (The only partition of 0 is [], having no number at all in it.)
  1 : 1;   (The only partition of 1 is [1], in which the number m=1 occurs 1 time.)
  2 : 0,0;   (Neither [2] nor [1,1] have some m occurring m times.)
  3 : 0,1,0;   ([3] and [1,1,1] have no m, but [1,2] has m=1 occurring m times.)
  4 : 0,1,1,0,0;   (Here [1,3] and [2,2] have m=1 resp. m=2 occurring m times.)
  5 : 0,1,0,0,2,0,0;   ([1,4] has m=1, [1,2,2] has m=1 and m=2 occurring m times.)
  6 : 0,1,0,0,0,1,0,0,1,0,0;
  7 : 0,1,0,0,0,1,1,1,0,0,1,0,1,0,0;
  (...)
Column 1 = (0,1,0,...) = A063524, characteristic function of {1}: The corresponding partition is [n], except for [] when n=0.
Column 2 = (0,1,1,1,...) = signum(n-2) = A057427(n-2), n >= 2: The corresponding partition is [1, n-1].
Column 3 = A063524(n-3) = A185014(n), characteristic function of {4}: The corresponding partition is [2, n-2] for n >= 4, and [1,1,1] for n = 3.
Column 4 = (0,...) = A000004(n-4), the zero function: The corresponding partition is [3, n-3] for n >= 6, and [1,1,2] for n = 4 and [1,1,3] for n = 5.
Row sums = A276428(n) = sum over all partitions of n of the number of distinct parts m of multiplicity m.

OEIS Wiki, Orderings of partitions

Cf. A036036 (list of partitions in Abramowitz & Stegun or graded reflected colexicographic order).
Cf. A000041 (partition numbers = row lengths).
Cf. A063524 (col.1: chi_{1}), A057427 (col.2: signum), A185014 (col.3: chi_{4}), A000004 (col.4: zero).
Cf. A276427 (frequency of 0, ..., max.value in each row), A276428 (row sums), A276429, A276434, A277101.
Cf. A328806 (row length of A276427(n) = 1 + largest value in row n).

apply( A328891_row(n, r=[])={forpart(p=n, my(s, c=1); for(i=1, #p, p[i]==if(i<#p, p[i+1]) && c++ && next; c==p[i] && s++; c=1); r=concat(r,s));r}, [0..12])

cp's OEIS Frontend

A276427 Irregular triangle read by rows: T(n,k) = number of partitions of n having k distinct parts i of multiplicity i; 0 <= k <= A328806(n)-1 = largest index of a nonzero value; n >= 0.

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A276434 Sum over all partitions of n of the number of distinct parts i of multiplicity i+1.

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A277101 Sum over all partitions of n of the number of distinct parts i of multiplicity i - 1.

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A276433 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i of multiplicity i+1 (n>=0).

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A277099 Number of partitions of n containing no part i of multiplicity i+1.

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A277100 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i (i>=2) of multiplicity i-1 (n>=0).

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Maple

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A277102 Number of partitions of n containing no part i of multiplicity i-1.

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