A261718 -id:A261718 - cp's OEIS frontend

A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 5, 0, 1, 5, 14, 22, 20, 7, 0, 1, 6, 20, 40, 51, 36, 11, 0, 1, 7, 27, 65, 105, 108, 65, 15, 0, 1, 8, 35, 98, 190, 252, 221, 110, 22, 0, 1, 9, 44, 140, 315, 506, 574, 429, 185, 30, 0, 1, 10, 54, 192, 490, 918, 1265, 1240, 810, 300, 42, 0

Offset: 0

Alois P. Heinz, Sep 09 2008

A(n,k) is also the number of partitions of n into parts of k kinds.
In general, column k > 0 is asymptotic to k^((k+1)/4) * exp(Pi*sqrt(2*k*n/3)) / (2^((3*k+5)/4) * 3^((k+1)/4) * n^((k+3)/4)) * (1 - (Pi*k^(3/2)/(24*sqrt(6)) + sqrt(3)*(k+1)*(k+3)/(8*Pi*sqrt(2*k))) / sqrt(n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017
When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field with k elements that contain an upper-triangular matrix. - Geoffrey Critzer, Nov 11 2022

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   2,   3,   4,   5, ...
  0,   2,   5,   9,  14,  20, ...
  0,   3,  10,  22,  40,  65, ...
  0,   5,  20,  51, 105, 190, ...
  0,   7,  36, 108, 252, 506, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-48. See Table 1.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Columns k=0-24 give: A000007, A000041, A000712, A000716, A023003, A023004, A023005, A023006, A023007, A023008, A023009, A023010, A005758, A023011, A023012, A023013, A023014, A023015, A023016, A023017, A023018, A023019, A023020, A023021, A006922.
Cf. A082556 (k=30), A082557 (k=32), A082558 (k=48), A082559 (k=64).
Rows n=0-4 give: A000012, A001477, A000096, A006503, A006504.
Main diagonal gives A008485.
Antidiagonal sums give A067687.
Cf. A060642, A122768, A246935, A255961, A261718.

# DedekindEta is defined in A000594.
A144064Column(k, len) = DedekindEta(len, -k)
for n in 0:8 A144064Column(n, 6) |> println end # Peter Luschny, Mar 10 2018

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->k)(n): seq(seq(A(n, d-n), n=0..d), d=0..14);

a[0, ] = 1; a[, 0] = 0; a[n_, k_] := SeriesCoefficient[ Product[1/(1 - x^j)^k, {j, 1, n}], {x, 0, n}]; Table[a[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2013 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d*p[d], {d, Divisors[j]} ]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[k&][n]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 30 2015, after Alois P. Heinz *)

Mat(apply( {A144064_col(k,nMax=9)=Col(1/eta('x+O('x^nMax))^k,nMax)}, [0..9])) \\ M. F. Hasler, Aug 04 2024

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

A(n,k) = Sum_{i=0..k} binomial(k,i) * A060642(n,k-i):

A074141 Sum of products of parts increased by 1 in all partitions of n.

Original entry on oeis.org

1, 2, 7, 18, 50, 118, 301, 684, 1621, 3620, 8193, 17846, 39359, 84198, 181313, 383208, 811546, 1695062, 3546634, 7341288, 15207022, 31261006, 64255264, 131317012, 268336125, 545858260, 1110092387, 2250057282, 4558875555, 9213251118, 18613373708, 37529713890

Offset: 0

Amarnath Murthy, Aug 28 2002

nonn

Replace each term in A036035 by the number of its divisors as in A074139; sequence gives sum of terms in the n-th row.

This is the sum of the number of submultisets of the multisets with n elements; a part of a partition is a frequency of such an element. - George Beck, Nov 01 2011

			The partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1, the corresponding products when parts are increased by 1 are 5,8,9,12,16 and their sum is a(4) = 50.

Alois P. Heinz, Table of n, a(n) for n = 0..3317

Cf. A006906, A036035, A074140, A256155.
Row sums of A074139 and of A079025 and of A079308 and of A238963.
Column k=2 of A261718.
Cf. A267008.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      2^n, b(n, i-1) +(1+i)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50); # Alois P. Heinz, Sep 07 2014

Table[Plus @@ Times @@@ (IntegerPartitions[n] + 1), {n, 0, 28}] (* T. D. Noe, Nov 01 2011 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, (1+i) * b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 08 2015, after Alois P. Heinz *)

S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

G.f.: 1/Product_{m>0} (1-(m+1)*x^m).
a(n) = 1/n*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d*(d+1)^(k/d).
a(n) = S(n,1), where S(n,m) = sum(k=m..n/2, (k+1)*S(n-k,k))+(n+1), S(n,n)=n+1, S(0,m)=1, S(n,m)=0 for nVladimir Kruchinin, Sep 07 2014
a(n) ~ c * 2^n, where c = Product_{k>=2} 1/(1-(k+1)/2^k) = 18.56314656361011472747535423226928404842588594722907068201... = A256155. - Vaclav Kotesovec, Sep 11 2014, updated May 10 2021

More terms from Alford Arnold, Sep 17 2002

More terms, better description and formulas from Vladeta Jovovic, Vladimir Baltic, Nov 28 2002

A261780 Number A(n,k) of compositions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 4, 0, 1, 4, 15, 24, 8, 0, 1, 5, 26, 73, 82, 16, 0, 1, 6, 40, 164, 354, 280, 32, 0, 1, 7, 57, 310, 1031, 1716, 956, 64, 0, 1, 8, 77, 524, 2395, 6480, 8318, 3264, 128, 0, 1, 9, 100, 819, 4803, 18501, 40728, 40320, 11144, 256, 0

Offset: 0

Alois P. Heinz, Aug 31 2015

Also the number of k-compositions of n: matrices with k rows of nonnegative integers with positive column sums and total element sum n.
A(2,2) = 7: (matrices and corresponding marked compositions are given)
  [1 1]  [0 0]  [1 0]  [0 1]  [1]   [2]   [0]
  [0 0]  [1 1]  [0 1]  [1 0]  [1]   [0]   [2]
  1a1a,  1b1b,  1a1b,  1b1a,  2ab,  2aa,  2bb.

			A(3,2) = 24: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a2aa, 1a2ab, 1a2bb, 1b2aa, 1b2ab, 1b2bb, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,      1, ...
  0,  1,   2,    3,     4,      5,      6, ...
  0,  2,   7,   15,    26,     40,     57, ...
  0,  4,  24,   73,   164,    310,    524, ...
  0,  8,  82,  354,  1031,   2395,   4803, ...
  0, 16, 280, 1716,  6480,  18501,  44022, ...
  0, 32, 956, 8318, 40728, 142920, 403495, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
E. Grazzini, E. Munarini, M. Poneti, S. Rinaldi, m-compositions and m-partitions: exhaustive generation and Gray code, Pure Math. Appl. 17 (2006), 111-121.
G. Louchard, Matrix Compositions: a Probabilistic analysis, Proc. GASCOM'08, Pure Mathematics and Applications, 19, 2-3, 127-146, 2008.
E. Munarini, M. Poneti, S. Rinaldi, Matrix compositions, Journal of Integer Sequences, Vol. 12 (2009), Article 09.4.8

Columns k=0-10 give: A000007, A011782, A003480, A145839, A145840, A145841, A161434, A261799, A261800, A261801, A261802.
Rows n=0-2 give: A000012, A001477, A005449.
Main diagonal gives A261783.
Cf. A261718 (same for partitions), A261781.

A:= proc(n, k) option remember; `if`(n=0, 1,
      add(A(n-j, k)*binomial(j+k-1, k-1), j=1..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[n_, k_] := SeriesCoefficient[(1-x)^k/(2*(1-x)^k-1), {x, 0, n}]; Table[ a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 07 2017 *)

G.f. of column k: (1-x)^k/(2*(1-x)^k-1).
A(n,k) = Sum_{i=0..k} C(k,i) * A261781(n,k-i).
A(n,k) = Sum_{j>=0} (1/2)^(j+1) * binomial(n-1+k*j,n). - Seiichi Manyama, Aug 06 2024

A261719 Number T(n,k) of partitions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order and all k letters occur at least once in the partition; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 3, 12, 10, 0, 5, 40, 81, 47, 0, 7, 104, 396, 544, 246, 0, 11, 279, 1751, 4232, 4350, 1602, 0, 15, 654, 6528, 25100, 44475, 36744, 11481, 0, 22, 1577, 23892, 136516, 369675, 512787, 352793, 95503, 0, 30, 3560, 80979, 666800, 2603670, 5413842, 6170486, 3641992, 871030

Offset: 0

Alois P. Heinz, Aug 29 2015

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.

			A(3,2) = 12: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a.
Triangle T(n,k) begins:
  1
  0,  1;
  0,  2,    3;
  0,  3,   12,    10;
  0,  5,   40,    81,     47;
  0,  7,  104,   396,    544,    246;
  0, 11,  279,  1751,   4232,   4350,   1602;
  0, 15,  654,  6528,  25100,  44475,  36744,  11481;
  0, 22, 1577, 23892, 136516, 369675, 512787, 352793, 95503;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A000041 (for n>0), A293366, A293367, A293368, A293369, A293370, A293371, A293372, A293373, A293374.
Row sums give A035341.
Main diagonal gives A005651.
T(2n,n) gives A261732.
Cf. A060642, A261718, A261781 (same for compositions).

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k]*Binomial[i + k - 1, k - 1]]]]; T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 21 2017, translated from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A261718(n,k-i).

A209668 a(n) = count of monomials, of degree k = n, in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

Original entry on oeis.org

1, 1, 7, 55, 631, 8001, 130453, 2323483, 48916087, 1129559068, 29442232007, 835245785452, 26113646252773, 880685234758941, 32191922753658129, 1259701078978200555, 52802268925363689079, 2352843030410455053891, 111343906794849929711260, 5567596199767400904172045

Offset: 0

Wouter Meeussen, Mar 11 2012

nonn

a(n) is the number of partitions of n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order. a(2) = 7: 2aa, 2ab, 2bb, 1a1a, 1a1b, 1b1a, 1b1b. - Alois P. Heinz, Aug 30 2015

Alois P. Heinz, Table of n, a(n) for n = 0..386
Wikipedia, Symmetric Polynomials

Cf. A209664-A209673.
Main diagonal of A209666 and A261718.
Cf. A261783.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 29 2015

h[n_, v_] := Tr@ Apply[Times, Table[Subscript[x, j], {j, v}]^# & /@ Compositions[n, v], {1}]; h[par_?PartitionQ, v_] := Times @@ (h[#, v] & /@ par); Tr /@ Table[(h[#, l] & /@ Partitions[l]) /. Subscript[x, _] -> 1, {l, 10}]
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n-i*j, i-1, k] * Binomial[i+k-1, k-1]^j, {j, 0, n/i}]]]; a[n_] := b[n, n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)

a(n) ~ c * n^n, where c = A247551 = Product_{k>=2} 1/(1-1/k!) = 2.529477472... . - Vaclav Kotesovec, Nov 15 2016

a(n) = [x^n] Product_{k>=1} 1 / (1 - binomial(k+n-1,n-1)*x^k). - Ilya Gutkovskiy, May 09 2021

a(0)=1 prepended and a(11)-a(19) added by Alois P. Heinz, Aug 29 2015

A261737 Number of partitions of n where each part i is marked with a word of length i over a ternary alphabet whose letters appear in alphabetical order.

Original entry on oeis.org

1, 3, 15, 55, 216, 729, 2621, 8535, 28689, 91749, 296538, 929712, 2939063, 9093255, 28257123, 86681608, 266368959, 811501848, 2475331535, 7505567037, 22772955015, 68828023329, 208079886258, 627418618533, 1892181244828, 5696253823476, 17149663331259

Offset: 0

Alois P. Heinz, Aug 30 2015

nonn

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

Column k=3 of A261718.

Cf. A293367.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(i+2, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]*Binomial[i + 2, 2]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 24 2018, translated from Maple *)

a(n) ~ c * 3^n, where c = Product_{k>=2} 1/(1 - (k+1)*(k+2)/(2*3^k)) = 6.84620607349852135789816336867607014231681538613599316638081993041973716978... . - Vaclav Kotesovec, Nov 15 2016, updated May 10 2021

G.f.: Product_{k>=1} 1 / (1 - binomial(k+2,2)*x^k). - Ilya Gutkovskiy, May 09 2021

A261738 Number of partitions of n where each part i is marked with a word of length i over a quaternary alphabet whose letters appear in alphabetical order.

Original entry on oeis.org

1, 4, 26, 124, 631, 2780, 12954, 55196, 241634, 1012196, 4280046, 17636252, 73157709, 298342936, 1220952044, 4947485904, 20079338277, 80987461760, 326986050564, 1314939934216, 5290893771329, 21236552526364, 85263892578686, 341801704446572, 1370448001291679

Offset: 0

Alois P. Heinz, Aug 30 2015

nonn

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

Column k=4 of A261718.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(i+3, 3))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i] Binomial[i + 3, 3]]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

a(n) ~ c * 4^n, where c = Product_{k>=2} 1/(1 - (k+1)*(k+2)*(k+3)/(3*2^(2*k+1))) = 4.90673361196637084263021203165784685586076564592828337755053385514766785... - Vaclav Kotesovec, Oct 11 2017, updated May 10 2021

G.f.: Product_{k>=1} 1 / (1 - binomial(k+3,3)*x^k). - Ilya Gutkovskiy, May 09 2021

A261739 Number of partitions of n where each part i is marked with a word of length i over a quinary alphabet whose letters appear in alphabetical order.

Original entry on oeis.org

1, 5, 40, 235, 1470, 8001, 45865, 241870, 1307055, 6783210, 35510502, 181665635, 934801705, 4741017595, 24118500815, 121693135003, 614889556920, 3091596201560, 15557885702390, 78054925105630, 391798489621630, 1963104427709830, 9838685572501515

Offset: 0

Alois P. Heinz, Aug 30 2015

nonn

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

Column k=5 of A261718.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(i+4, 4))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]*Binomial[i + 4, 4]]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

a(n) ~ c * 5^n, where c = Product_{k>=2} 1/(1 - (k+1)*(k+2)*(k+3)*(k+4)/(24*5^k)) = 4.1548340497015786311470026968208254860294132084317763408428889184148319... - Vaclav Kotesovec, Oct 11 2017, updated May 10 2021

G.f.: Product_{k>=1} 1 / (1 - binomial(k+4,4)*x^k). - Ilya Gutkovskiy, May 09 2021

A261740 Number of partitions of n where each part i is marked with a word of length i over a senary alphabet whose letters appear in alphabetical order.

Original entry on oeis.org

1, 6, 57, 398, 2955, 19158, 130453, 820554, 5280204, 32711022, 204324819, 1249546656, 7682267669, 46625705988, 283766862009, 1714704081724, 10374896682273, 62511439251768, 376943252871343, 2267304042230202, 13643684237963994, 81983795625450144

Offset: 0

Alois P. Heinz, Aug 30 2015

nonn

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

Column k=6 of A261718.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(i+5, 5))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

a(n) ~ c * 6^n, where c = Product_{k>=2} 1/(1 - binomial(k+5,5)/6^k) = 3.760725122262068858184072984846959348360490081749654779894152320389687335... - Vaclav Kotesovec, Oct 11 2017, updated May 10 2021

G.f.: Product_{k>=1} 1 / (1 - binomial(k+5,5)*x^k). - Ilya Gutkovskiy, May 09 2021

A261741 Number of partitions of n where each part i is marked with a word of length i over a septenary alphabet whose letters appear in alphabetical order.

Original entry on oeis.org

1, 7, 77, 623, 5355, 40299, 317905, 2323483, 17353028, 124991685, 907465307, 6458846989, 46199021001, 326573565143, 2314422214435, 16296707707077, 114891467946017, 806991845455033, 5672334432498356, 39785054428093380, 279156880971492454, 1956352659297436368

Offset: 0

Alois P. Heinz, Aug 30 2015

nonn

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

Column k=7 of A261718.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(i+6, 6))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

a(n) ~ c * 7^n, where c = Product_{k>=2} 1/(1 - binomial(k+6,6)/7^k) = 3.519268129363442517546929108933080435102442778133731795486515352... - Vaclav Kotesovec, Oct 11 2017, updated May 10 2021

G.f.: Product_{k>=1} 1 / (1 - binomial(k+6,6)*x^k). - Ilya Gutkovskiy, May 10 2021

cp's OEIS Frontend

A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Julia

Maple

Mathematica

PARI

Formula

A074141 Sum of products of parts increased by 1 in all partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

Extensions

A261780 Number A(n,k) of compositions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A261719 Number T(n,k) of partitions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order and all k letters occur at least once in the partition; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A209668 a(n) = count of monomials, of degree k = n, in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A261737 Number of partitions of n where each part i is marked with a word of length i over a ternary alphabet whose letters appear in alphabetical order.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A261738 Number of partitions of n where each part i is marked with a word of length i over a quaternary alphabet whose letters appear in alphabetical order.

Views

Author

Keywords

Links