A000065 -id:A000065 - cp's OEIS frontend

A304966 Expansion of Product_{k>=1} 1/(1 - x^k)^(p(k)-1), where p(k) = number of partitions of k (A000041).

1, 0, 1, 2, 5, 8, 18, 30, 61, 107, 203, 358, 663, 1162, 2093, 3666, 6481, 11258, 19652, 33874, 58464, 100046, 171032, 290563, 492745, 831393, 1399655, 2346707, 3924873, 6541472, 10875575, 18025629, 29804125, 49143254, 80841455, 132651457, 217179366, 354745107, 578215807

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Euler transform of A000065.

Convolution of the sequences A001970 and A010815.

N. J. A. Sloane, Transforms
Index entries for sequences related to partitions

Cf. A000041, A000065, A000219, A001383, A001970, A010815, A052847.

with(combinat): with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d)-d, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 22 2018

nmax = 38; CoefficientList[Series[Product[1/(1 - x^k)^(PartitionsP[k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (PartitionsP[d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 38}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A000065(k).

A327639 Number T(n,k) of proper k-times partitions of n; triangle T(n,k), n >= 0, 0 <= k <= max(0,n-1), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 3, 1, 6, 15, 16, 6, 1, 10, 45, 88, 76, 24, 1, 14, 93, 282, 420, 302, 84, 1, 21, 223, 1052, 2489, 3112, 1970, 498, 1, 29, 444, 2950, 9865, 18123, 18618, 10046, 2220, 1, 41, 944, 9030, 42787, 112669, 173338, 155160, 74938, 15108

Offset: 0

Alois P. Heinz, Sep 20 2019

In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.
T(n,k) is defined for all n>=0 and k>=0.  The triangle displays only positive terms.  All other terms are zero.
Row n is the inverse binomial transform of the n-th row of array A323718.

			T(4,0) = 1:  4
T(4,1) = 4:     T(4,2) = 6:          T(4,3) = 3:
  4-> 31          4-> 31 -> 211        4-> 31 -> 211 -> 1111
  4-> 22          4-> 31 -> 1111       4-> 22 -> 112 -> 1111
  4-> 211         4-> 22 -> 112        4-> 22 -> 211 -> 1111
  4-> 1111        4-> 22 -> 211
                  4-> 22 -> 1111
                  4-> 211-> 1111
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  2,   1;
  1,  4,   6,    3;
  1,  6,  15,   16,     6;
  1, 10,  45,   88,    76,     24;
  1, 14,  93,  282,   420,    302,     84;
  1, 21, 223, 1052,  2489,   3112,   1970,    498;
  1, 29, 444, 2950,  9865,  18123,  18618,  10046,  2220;
  1, 41, 944, 9030, 42787, 112669, 173338, 155160, 74938, 15108;
  ...

Alois P. Heinz, Rows n = 0..170, flattened
Wikipedia, Iverson bracket
Wikipedia, Partition (number theory)

Columns k=0-2 give A000012, A000065, A327769.
Row sums give A327644.
T(2n,n) gives A327645.
Cf. A323718, A327631, A327643, A327646.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
      b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..max(0, n-1)), n=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, Max[0, n - 1] }] // Flatten (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A323718(n,i).
T(n,n-1) = A327631(n,n-1)/n = A327643(n) for n >= 1.
Sum_{k=1..n-1} k * T(n,k) = A327646(n).
Sum_{k=0..max(0,n-1)} (-1)^k * T(n,k) = [n<2], where [] is an Iverson bracket.

A167934 a(n) = A000041(n) - A032741(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 8, 14, 19, 28, 39, 55, 72, 100, 132, 173, 227, 296, 380, 489, 622, 789, 999, 1254, 1568, 1956, 2433, 3007, 3713, 4564, 5597, 6841, 8344, 10140, 12307, 14880, 17969, 21636, 26012, 31182, 37331, 44582, 53167, 63260, 75170

Offset: 0

Omar E. Pol, Nov 16 2009

nonn

a(n) is also the number of partitions of n whose parts are not all equal, (including however the partition with a single part of size n). Note that the number of partitions of n whose parts are all equal gives the number of divisors of n, for n>0. (See also A144300.)

			The partitions of n = 6 are:
6 ....................... All parts are equal, but included .. (1).
5 + 1 ................... All parts are not equal ............ (2).
4 + 2 ................... All parts are not equal ............ (3).
4 + 1 + 1 ............... All parts are not equal ............ (4).
3 + 3 ................... All parts are equal, not included.
3 + 2 + 1 ............... All parts are not equal ............ (5).
3 + 1 + 1 + 1 ........... All parts are not equal ............ (6).
2 + 2 + 2 ............... All parts are equal, not included.
2 + 2 + 1 + 1 ........... All parts are not equal ............ (7).
2 + 1 + 1 + 1 + 1 ....... All parts are not equal ............ (8).
1 + 1 + 1 + 1 + 1 + 1 ... All parts are equal, not included.
Then a(6) = 8.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Omar E. Pol, Illustration of the shell model of partitions (2D and 3D view)
Omar E. Pol, Illustration of the shell model of partitions (2D view)
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A144300, A135010, A138121, A167930, A167932, A167935.

b:= proc(n, i, k) option remember;
      if n<0 then 0
    elif n=0 then `if`(k=0, 1, 0)
    elif i=0 then 0
    else b(n, i-1, k)+
         b(n-i, i, `if`(k<0, i, `if`(k<>i, 0, k)))
      fi
    end:
a:= n-> 1 +b(n, n-1, -1):
seq(a(n), n=0..50);  #  Alois P. Heinz, Dec 01 2010

a[0] = 1; a[n_] := PartitionsP[n] - DivisorSigma[0, n] + 1; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 08 2016 *)

a(n) = A000041(n) - A032741(n).

More terms from Alois P. Heinz, Dec 01 2010

A058682 a(n) = p(0) + p(1) + ... + p(n) - n - 1, where p = partition numbers, A000041.

Original entry on oeis.org

0, 0, 1, 3, 7, 13, 23, 37, 58, 87, 128, 183, 259, 359, 493, 668, 898, 1194, 1578, 2067, 2693, 3484, 4485, 5739, 7313, 9270, 11705, 14714, 18431, 22995, 28598, 35439, 43787, 53929, 66238, 81120, 99096, 120732, 146746, 177930, 215267, 259849, 313022, 376282

Offset: 0

N. J. A. Sloane, Dec 30 2000

nonn

Number of non-isomorphic rank-2 matroids over S_n.

Starting (1, 3, 7, 13, ...) = row sums of triangle A171239. - Gary W. Adamson, Dec 05 2009

Acketa, Dragan M. "On the enumeration of matroids of rank-2." Zbornik radova Prirodnomatematickog fakulteta-Univerzitet u Novom Sadu 8 (1978): 83-90. - N. J. A. Sloane, Dec 04 2022

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 501 terms from Muniru A Asiru)
W. M. B. Dukes, Tables of matroids.
W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
Markus Kirchweger, Manfred Scheucher, and Stefan Szeider, A SAT Attack on Rota's Basis Conjecture, Leibniz International Proceedings in Informatics (LIPIcs 2022) Vol. 236, 4:1-4:18.
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, arXiv:math/0702316 [math.CO], 2007 (see p. 7).
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
Index entries for sequences related to matroids

Column k=2 of A053534.
Cf. A000041, A000065 (first differences), A000070.
Cf. A171239. - Gary W. Adamson, Dec 05 2009

List([1..41],n->Sum([1..n-1],k->NrPartitions(k)-1)); # Muniru A Asiru, Aug 10 2018

a:= proc(n) option remember; `if`(n<2, 0,
      combinat[numbpart](n)+a(n-1)-1)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 10 2019

With[{s = PartitionsP /@ Range[0, 40]}, MapIndexed[Total@ Take[s, First@ #2] - First@ #2 &, s]] (* Michael De Vlieger, Sep 04 2017 *)
With[{nn=50},#[[2]]-#[[1]]&/@Thread[{Range[0,nn],Accumulate[PartitionsP[Range[0,nn]]]}]]-1 (* Harvey P. Dale, Sep 05 2023 *)

G.f.: -1/(1 - x)^2 + (1/(1 - x))*Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Aug 10 2018

Name clarified by Ilya Gutkovskiy, Aug 10 2018

A058716 Triangle T(n,k) giving number of nonisomorphic loopless matroids of rank k on n labeled points (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 6, 9, 4, 1, 0, 1, 10, 25, 18, 5, 1, 0, 1, 14, 70, 85, 31, 6, 1, 0, 1, 21, 217, 832, 288, 51, 7, 1

Offset: 0

N. J. A. Sloane, Dec 31 2000

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  2,   1;
  0, 1,  4,   3,   1;
  0, 1,  6,   9,   4,   1;
  0, 1, 10,  25,  18,   5,  1;
  0, 1, 14,  70,  85,  31,  6, 1;
  0, 1, 21, 217, 832, 288, 51, 7, 1;
  ...

W. M. B. Dukes, Tables of matroids.
W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
Index entries for sequences related to matroids

Cf. A058717 (same except for border), A058710, A058711. Row sums give A058718. Diagonals give A000065, A058719.

Corrected and extended by Jean-François Alcover, Oct 21 2013

Reverted to original data by Sean A. Irvine, Aug 16 2022

A096806 Triangle, read by rows, such that the binomial transform of the n-th row lists the m-dimensional partitions of n, for n>=1 and m>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 11, 7, 1, 1, 10, 27, 28, 11, 1, 1, 14, 57, 93, 64, 16, 1, 1, 21, 117, 269, 282, 131, 22, 1, 1, 29, 223, 707, 1062, 766, 244, 29, 1, 1, 41, 417, 1747, 3565, 3681, 1871, 421, 37, 1, 1, 55, 748, 4090, 10999, 15489, 11400, 4152, 683, 46, 1, 1, 76

Offset: 1

Paul D. Hanna, Jul 19 2004

The n-th row equals the inverse binomial transform of n-th column of square array A096751, for n>=1. The zero-dimensional partition of n is taken to be 1 for all n.

			The number of m-dimensional partitions of 5, for m>=0, is given by the binomial transform of the 5th row:
BINOMIAL([1,6,11,7,1]) = [1,7,24,59,120,216,357,554,819,1165,...] = A008779.
Rows begin:
  [1],
  [1,  1],
  [1,  2,   1],
  [1,  4,   4,    1],
  [1,  6,  11,    7,     1],
  [1, 10,  27,   28,    11,     1],
  [1, 14,  57,   93,    64,    16,      1],
  [1, 21, 117,  269,   282,   131,     22,      1],
  [1, 29, 223,  707,  1062,   766,    244,     29,     1],
  [1, 41, 417, 1747,  3565,  3681,   1871,    421,    37,     1],
  [1, 55, 748, 4090, 10999, 15489,  11400,   4152,   683,    46,    1],
  [1, 76,1326, 9219, 31828, 58975,  59433,  31802,  8483,  1054,   56,   1],
  [1,100,2284,20095, 87490,207735, 276230, 204072, 80664, 16162, 1561,  67, 1],
  [1,134,3898,42707,230737,687665,1173533,1148939,632478,188077,29031,2234,79,1],
  ...
The inverse binomial transform of the diagonals of this triangle begin:
  [1],
  [1, 1,  1],
  [1, 3,  4,   6,  3],
  [1, 5, 16,  29,  49,   45,   15],
  [1, 9, 38, 127, 289,  540,  660,   420, 105],
  [1,13, 90, 397,1384, 3633, 7506, 10920,9765,4725,945],
  [1,20,182,1140,5266,19324,55645,125447,  ? ,  ? , ?  ,62370,10395],
  ...

S. Govindarajan, Notes on higher-dimensional partitions, arXiv:1203.4419 [math.CO], 2012.

Cf. A096751, A096807 (row sums), A000065 (column k=1?), A008778 (bin trans 4th row), A042984 (bin trans 6th row)

Cf. A119271.

T(n, 0)=T(n, n-1)=1, T(n, 1)=A000041(n)-1, T(n, n-2)=(n-1)*(n-2)/2+1, for n>=1.

A104382 Triangle, read by rows, where T(n,k) equals number of distinct partitions of triangular number n*(n+1)/2 into k different summands for n>=k>=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 12, 6, 1, 1, 10, 27, 27, 10, 1, 1, 13, 52, 84, 57, 14, 1, 1, 17, 91, 206, 221, 110, 21, 1, 1, 22, 147, 441, 674, 532, 201, 29, 1, 1, 27, 225, 864, 1747, 1945, 1175, 352, 41, 1, 1, 32, 331, 1575, 4033, 5942, 5102, 2462, 598, 55, 1, 1, 38, 469

Offset: 1

Paul D. Hanna, Mar 04 2005

Secondary diagonal equals partitions of n - 1 (A000065).
Third diagonal is A104384.
Third column is A104385.
Row sums are A104383 where limit_{n --> inf} A104383(n+1)/A104383(n) = exp(sqrt(Pi^2/6)) = 3.605822247984...

			Rows begin:
1;
1, 1;
1, 2, 1;
1, 4, 4, 1;
1, 7, 12, 6, 1;
1, 10, 27, 27, 10, 1;
1, 13, 52, 84, 57, 14, 1;
1, 17, 91, 206, 221, 110, 21, 1;
1, 22, 147, 441, 674, 532, 201, 29, 1;
1, 27, 225, 864, 1747, 1945, 1175, 352, 41, 1;
1, 32, 331, 1575, 4033, 5942, 5102, 2462, 598, 55, 1; ...

Abramowitz, M. and Stegun, I. A. (Editors). "Partitions into Distinct Parts." S24.2.2 in Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, pp. 825-826, 1972.

Alois P. Heinz, Rows n = 1..55, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Weisstein's World of Mathematics, Partition Function Q.

Cf. A008289, A000009, A000065, A104383, A104384, A104385.

PARI
```
T(n,k)=if(n
				
```

T(n, 1) = T(n, n) = 1.
T(n, n-1) = A000065(n).
T(n, 2) = [(n*(n+1)/2-1)/2].
From Álvar Ibeas, Jul 23 2020: (Start)
T(n, k) = A008284((n-k+1)*(n+k)/2, k).
T(n, k) = A026820((n-k)*(n+k+1)/2, k), with A026820(0, k) = 1. (End)

A198381 Total number of parts greater than 1 in all partitions of n minus the number of partitions of n into parts each less than n.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 6, 10, 20, 32, 54, 81, 128, 184, 273, 385, 549, 754, 1048, 1412, 1917, 2547, 3392, 4444, 5837, 7556, 9791, 12553, 16086, 20429, 25935, 32665, 41108, 51404, 64190, 79721, 98882, 122043, 150417, 184618, 226239

Offset: 0

Omar E. Pol, Oct 27 2011

nonn

Also partial sums of A182699. Total number of emergent parts in all partitions of the numbers <= n.

Also total number of parts of all regions of n that do not contain 1 as a part (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012

Cf. A000041, A000065, A000070, A006128, A026905, A093694, A096541, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196930, A196931.

a(n) = A096541(n) - A000065(n) = 1 + A096541(n) - A000041(n) = 1 + A006128(n) - A000070(n).

a(n) = A006128(n) - A026905(n), n >= 1.

A000147 Number of trees of diameter 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 7, 14, 32, 58, 110, 187, 322, 519, 839, 1302, 2015, 3032, 4542, 6668, 9738, 14006, 20036, 28324, 39830, 55473, 76875, 105692, 144629, 196585, 266038, 357952, 479664, 639519, 849425, 1123191, 1479972, 1942284, 2540674, 3311415

Offset: 1

N. J. A. Sloane

nonn

A tree of diameter 5 is formed from two rooted trees of height 2, with their roots joined. - Franklin T. Adams-Watters, Jan 13 2006

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
Index entries for sequences related to trees

Cf. A034853, A000251 (diameter 6).

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0,
     add(binomial(b((i-1)$2, k-1)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
g:= n-> b((n-1)$2, 2) -b((n-1)$2, 1):
a:= n-> (add(g(i)*g(n-i), i=0..n)+`if`(n::even, g(n/2), 0))/2:
seq(a(n), n=1..45);  # Alois P. Heinz, Feb 09 2016

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1 || k<1, 0, Sum[Binomial[ b[i-1, i-1, k-1]+j-1, j]*b[n-i*j, i-1, k], {j, 0, n/i}]]]; g[n_] := b[n-1, n-1, 2] - b[n-1, n-1, 1]; a[n_] := (Sum[g[i]*g[n-i], {i, 0, n}] + If[EvenQ[n], g[n/2], 0])/2; Table[a[n], {n, 1, 45}] (* Jean-François Alcover, Feb 17 2016, after Alois P. Heinz *)

If n odd, a(n) = Sum_{k=1..(n-1)/2} b(k)*b(n-k); if n even, a(n) = (Sum_{k=1..n/2-1} b(k)*b(n-k)) + C(b(n/2)+1, 2), where b(n) = P(n-1)-1 = A000065(n-1). - Franklin T. Adams-Watters, Jan 13 2006

More terms from Franklin T. Adams-Watters, Jan 13 2006

A249543 Square array T(m,n) of integer partitions with m addends n+1, read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 9, 7, 6, 20, 26, 15, 10, 40, 72, 68, 30, 14, 75, 171, 220, 159, 56, 21, 133, 379, 614, 603, 352, 101, 29, 229, 786, 1559, 1928, 1525, 732, 176, 41, 383, 1568, 3700, 5564, 5534, 3618, 1465, 297

Offset: 1

Tilman Piesk, Oct 31 2014

T(m,n) is the integer partition with m times the addend n+1 (and no other non-one addends) given as index number of A194602.
The entries in the array A249544 are also in the sequence A194602. This array T contains the index numbers of A194602 corresponding to the entries of that array: A194602(T(m,n)) = A249544(m,n).
Row 1 is A000065, column 1 is A058695 (both with shifted index).

			T(5,2) = 159.
A194602(159) = 14043. (So A249544(5,2) = 14043.)
14043 in binary is 11011011011011. That corresponds to the integer partition with 5 times the addend 3. (See row 159 in "Table for A194602" link.)
Array begins:
   n    1    2    3    4    5    6    7   8   9
m
1       1    2    4    6   10   14   21  29  41
2       3    9   20   40   75  133  229 383
3       7   26   72  171  379  786 1568
4      15   68  220  614 1559 3700
5      30  159  603 1928 5564
6      56  352 1525 5534
7     101  732 3618
8     176 1465
9     297

Tilman Piesk, First 113 rows of the triangle, flattened
Tilman Piesk, Table for A194602
Li-yao Xia, Identities for A194602

Cf. A194602, A249544, A000065, A058695, A000041.

A194602(T(m,n)) =  A249544(m,n).
T(1,n) = A000065(n+1) = p(n+1) - 1.
T(2,n) = p(2*(n+1)) - 2.
T(3,n) = p(3*(n+1)) - floor((n+1)/2) - 3.
T(m,1) = A058695(m-1) = p(2n-1).
p is the sequence of partition numbers A000041. (See "Identities for A194602" link.)

cp's OEIS Frontend

A304966 Expansion of Product_{k>=1} 1/(1 - x^k)^(p(k)-1), where p(k) = number of partitions of k (A000041).

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A327639 Number T(n,k) of proper k-times partitions of n; triangle T(n,k), n >= 0, 0 <= k <= max(0,n-1), read by rows.

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A167934 a(n) = A000041(n) - A032741(n).

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Maple

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A058682 a(n) = p(0) + p(1) + ... + p(n) - n - 1, where p = partition numbers, A000041.

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GAP

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A058716 Triangle T(n,k) giving number of nonisomorphic loopless matroids of rank k on n labeled points (n >= 0, 0 <= k <= n).

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A096806 Triangle, read by rows, such that the binomial transform of the n-th row lists the m-dimensional partitions of n, for n>=1 and m>=0.

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A104382 Triangle, read by rows, where T(n,k) equals number of distinct partitions of triangular number n*(n+1)/2 into k different summands for n>=k>=1.

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PARI

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