cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A195820 Total number of smallest parts in all partitions of n that do not contain 1 as a part.

Original entry on oeis.org

0, 1, 1, 3, 2, 7, 5, 12, 13, 22, 22, 43, 43, 67, 81, 117, 133, 195, 223, 312, 373, 492, 584, 782, 925, 1190, 1433, 1820, 2170, 2748, 3268, 4075, 4872, 5997, 7150, 8781, 10420, 12669, 15055, 18198, 21535, 25925, 30602, 36624, 43201, 51428, 60478, 71802, 84215
Offset: 1

Views

Author

Omar E. Pol, Oct 19 2011

Keywords

Comments

Total number of smallest parts in all partitions of the head of the last section of the set of partitions of n.

Examples

			For n = 8 the seven partitions of 8 that do not contain 1 as a part are:
.  (8)
.  (4) + (4)
.   5  + (3)
.   6  + (2)
.   3  +  3  + (2)
.   4  + (2) + (2)
.  (2) + (2) + (2) + (2)
Note that in every partition the smallest parts are shown between parentheses. The total number of smallest parts is 1+2+1+1+1+2+4 = 12, so a(8) = 12.

Links

Crossrefs

Cf. A000041, A000070, A002865, A092269, A135010, A138121, A138135, A138137, A182984.

Programs

  • Maple
    b:= proc(n, i) option remember;
      `if`(n=0 or i<2, 0, b(n, i-1)+
       add(`if`(n=i*j, j, b(n-i*j, i-1)), j=1..n/i))
    end:
a:= n-> b(n, n):
seq(a(n), n=1..60); # Alois P. Heinz, Apr 09 2012
  • Mathematica
    Table[s = Select[IntegerPartitions[n], ! MemberQ[#, 1] &]; Plus @@ Table[Count[x, Min[x]], {x, s}], {n, 50}] (* T. D. Noe, Oct 19 2011 *)
b[n_, i_] := b[n, i] = If[n==0 || i<2, 0, b[n, i-1] + Sum[If[n== i*j, j, b[n-i*j, i-1]], {j, 1, n/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)
  • Sage
    def A195820(n):
    return sum(list(p).count(min(p)) for p in Partitions(n,min_part=2))
# D. S. McNeil, Oct 19 2011

Formula

a(n) = A092269(n) - A000070(n-1).
G.f.: Sum_{i>=2} x^i/(1 - x^i) * Product_{j>=i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 03 2017
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n) * (1 - (72 + 5*Pi^2)*sqrt(6) / (144*Pi*sqrt(n))). - Vaclav Kotesovec, Jul 31 2017

Extensions

More terms from D. S. McNeil, Oct 19 2011

A182977 Total number of parts that are neither the smallest part nor the largest part in all partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 6, 12, 22, 39, 66, 103, 159, 243, 352, 510, 721, 1011, 1391, 1903, 2557, 3436, 4549, 5999, 7824, 10187, 13132, 16886, 21544, 27414, 34657, 43703, 54797, 68558, 85328, 105963, 131028, 161664, 198710
Offset: 0

Views

Author

Omar E. Pol, Jul 17 2011

Keywords

Examples

			For n = 6 the partitions of 6 are
6
5 + 1
4 + 2
4 + 1 + 1
3 + 3
3 + (2) + 1 .......... the "2" is the part that counts.
3 + 1 + 1 + 1
2 + 2 + 2
2 + 2 + 1 + 1
2 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1
There is only one part which is neither the smallest part nor the largest part in all partitions of 6, so a(6) = 1.

Links

Crossrefs

Cf. A006128, A046746, A092269, A116686, A182978, A182984, A265249.

Programs

  • Maple
    g := add(add((add(x^(i+j+k)/(1-x^k), k = i+1 .. j-1))/(mul(1-x^k, k = i .. j)), j = i+1 .. 80), i = 1 .. 80): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 45); # Emeric Deutsch, Dec 25 2015

Formula

a(n) = A006128(n) - A182978(n).
G.f.: g(x) = Sum_{i>=1} Sum_{j>=i+1} (Sum_{k=i+1..j-1} x^{i+j+k}/(1-x^k)/Product_{k=i..j}(1-x^k)). - Emeric Deutsch, Dec 25 2015
a(n) = Sum_{k>=0} k*A265249(n,k). - Emeric Deutsch, Dec 25 2015

Extensions

a(12) corrected and more terms a(13)-a(40) from David Scambler, Jul 18 2011

A264402 Triangle read by rows: T(n,k) is the number of partitions of n that have k parts larger than the smallest part (n>=1, k>=0).

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 2, 4, 1, 4, 5, 2, 2, 8, 4, 1, 4, 9, 7, 2, 3, 12, 10, 4, 1, 4, 14, 15, 7, 2, 2, 17, 20, 12, 4, 1, 6, 18, 27, 17, 7, 2, 2, 23, 33, 26, 12, 4, 1, 4, 24, 44, 35, 19, 7, 2, 4, 27, 51, 49, 28, 12, 4, 1, 5, 30, 64, 63, 41, 19, 7, 2, 2
Offset: 1

Views

Author

Emeric Deutsch, Nov 21 2015

Keywords

Comments

T(n,k) = number of partitions of n in which the 2nd largest part is k (0 if all parts are equal). Example: T(7,2) = 4 because we have [3,2,1,1], [3,2,2], [4,2,1], and [5,2].
The fact that the above two statistics (in Name and in 1st Comment) have the same distribution follows at once by conjugation. - Emeric Deutsch, Dec 11 2015
Row sums yield the partition numbers (A000041).
T(n,0) = A000005(n) = number of divisors of n.
Sum_{k>=0} k*T(n,k) = A182984(n).

Examples

			T(7,2) = 4 because we have [2,2,1,1,1], [3,2,1,1], [3,3,1], and [4,2,1].
Triangle starts:
1;
2;
2,1;
3,2;
2,4,1;
4,5,2;
2,8,4,1;

Links

Crossrefs

Cf. A000005, A000041, A116685, A182984, A002541.

Programs

  • Maple
    g := sum(x^i/((1-x^i)*(product(1-t*x^j, j = i+1 .. 100))), i = 1 .. 100): gser := simplify(series(g, x = 0, 30)): for n to 27 do P[n] := sort(coeff(gser, x, n)) end do: for n to 27 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, 0, b(n, i-1) +add((p->[0, p[1]+
       expand(p[2]*x^j)])(b(n-i*j, i-1)) , j=1..n/i)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)[2]):
seq(T(n), n=1..20);  # Alois P. Heinz, Nov 29 2015
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, b[n, i-1] + Sum[ Function[p, {0, p[[1]] + Expand[p[[2]]*x^j]}][b[n-i*j, i-1]], {j, 1, n/i} ]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n][[2]]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)

Formula

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

A182978 Total number of parts that are the smallest part or the largest part in all partitions of n.

Original entry on oeis.org

1, 3, 6, 12, 20, 34, 52, 80, 116, 170, 236, 333, 453, 621, 825, 1111, 1455, 1923, 2487, 3239, 4149, 5342, 6770, 8625, 10852, 13698, 17107, 21413, 26567, 33019, 40721, 50270, 61663, 75665, 92318, 112686, 136849, 166173, 200923, 242836
Offset: 1

Views

Author

Omar E. Pol, Jul 17 2011

Keywords

Examples

			For n = 6 the partitions of 6 are
6
5 + 1
4 + 2
4 + 1 + 1
3 + 3
3 + (2) + 1 .... the "2" is the part that does not count.
3 + 1 + 1 + 1
2 + 2 + 2
2 + 2 + 1 + 1
2 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1
The total number of parts in all partitions of 6 is equal to 35. All parts are the smallest part or the largest part, except the "2" in the partition (3 + 2 + 1), so a(6) = 35 - 1 = 34.

Links

Crossrefs

Cf. A006128, A046746, A092269, A116686, A182977, A182984.

Programs

  • Maple
    l:= proc(n, i) option remember; `if`(n=i, n, 0)+
      `if`(i<1, 0, l(n, i-1) +`if`(n l(n, n) +s(n, n) -numtheory[sigma](n):
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 17 2013
  • Mathematica
    l[n_, i_] := l[n, i] = If[n==i, n, 0] + If[i<1, 0, l[n, i-1] + If[nJean-François Alcover, Nov 03 2015, after Alois P. Heinz *)

Formula

a(n) = A006128(n) - A182977(n).

Extensions

a(12) corrected and more terms a(13)-a(40) from David Scambler, Jul 18 2011

A195821 Total number of parts that are not the smallest part in all partitions of n that do not contain 1 as a part.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 5, 7, 12, 19, 25, 37, 56, 72, 102, 138, 187, 246, 330, 422, 563, 721, 931, 1177, 1523, 1903, 2421, 3020, 3797, 4700, 5875, 7218, 8956, 10954, 13474, 16401, 20083, 24316, 29576, 35685, 43179, 51870, 62490, 74757, 89666, 106927, 127687
Offset: 1

Views

Author

Omar E. Pol, Oct 19 2011

Keywords

Comments

Total number of parts that are not the smallest part in all partitions of the head of the last section of the set of partitions of n. For more information see A195820.

Examples

			For n = 8 the seven partitions of 8 that do not contain 1 as a part are:
.   8
.   4  +  4
.  (5) +  3
.  (6) +  2
.  (3) + (3) +  2
.  (4) +  2  +  2
.   2  +  2  +  2  +  2
Note that in every partition the parts that are not the smallest part are shown between parentheses. The total number of parts that are not the smallest part is 0+0+1+1+2+1+0 = 5, so a(8) = 5.

Links

Crossrefs

Cf. A000041, A000070, A002865, A092269, A135010, A138121, A138135, A138137, A182984, A195820.

Formula

a(n) = A138135(n) - A195820(n) = A138137(n) - A195820(n) - A000041(n-1).
Showing 1-5 of 5 results.

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