A059481 -id:A059481 - cp's OEIS frontend

A027555 Triangle of binomial coefficients C(-n,k).

1, 1, -1, 1, -2, 3, 1, -3, 6, -10, 1, -4, 10, -20, 35, 1, -5, 15, -35, 70, -126, 1, -6, 21, -56, 126, -252, 462, 1, -7, 28, -84, 210, -462, 924, -1716, 1, -8, 36, -120, 330, -792, 1716, -3432, 6435, 1, -9, 45, -165, 495, -1287, 3003, -6435, 12870, -24310, 1, -10, 55, -220, 715, -2002, 5005, -11440, 24310, -48620, 92378

Offset: 0

N. J. A. Sloane, Olivier Gérard

			Triangle starts:
  1;
  1, -1;
  1, -2,  3;
  1, -3,  6, -10;
  1, -4, 10, -20, 35;
  1, -5, 15, -35, 70, -126;
  ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 164.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 2.

T. D. Noe, Rows n = 0..50 of triangle, flattened

For the unsigned triangle see A059481.

/* As triangle */ [[Binomial(-n, k): k in [0..n]]: n in [0..11]]; // G. C. Greubel, Nov 21 2017

A027555 := proc(n,k)
    (-1)^k*binomial(n+k-1,k) ;
end proc:
seq(seq(A027555(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Feb 06 2015

Flatten[Table[Binomial[-n,k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Apr 30 2012 *)

T(n,k)=binomial(-n,k) \\ Charles R Greathouse IV, Feb 06 2017

def T(n,k):
    return (-1)^k * rising_factorial(n, k) // factorial(k)
for n in range(9):
    print([T(n, k) for k in range(n+1)])  # Peter Luschny, Nov 24 2023

T(n,k) = binomial(-n,k) = (-1)^k*binomial(n+k-1,k). - R. J. Mathar, Feb 06 2015

T(n, k) = (-1)^k * RisingFactorial(n, k) / k!. - Peter Luschny, Nov 24 2023

A343658 Array read by antidiagonals where A(n,k) is the number of ways to choose a multiset of k divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 6, 2, 1, 1, 6, 5, 10, 3, 4, 1, 1, 7, 6, 15, 4, 10, 2, 1, 1, 8, 7, 21, 5, 20, 3, 4, 1, 1, 9, 8, 28, 6, 35, 4, 10, 3, 1, 1, 10, 9, 36, 7, 56, 5, 20, 6, 4, 1, 1, 11, 10, 45, 8, 84, 6, 35, 10, 10, 2, 1

Offset: 1

Gus Wiseman, Apr 29 2021

First differs from A343656 at A(4,2) = 6, A343656(4,2) = 5.

As a triangle, T(n,k) = number of ways to choose a multiset of n - k divisors of k.

			Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=1:  1   1   1   1   1   1   1   1   1
  n=2:  1   2   3   4   5   6   7   8   9
  n=3:  1   2   3   4   5   6   7   8   9
  n=4:  1   3   6  10  15  21  28  36  45
  n=5:  1   2   3   4   5   6   7   8   9
  n=6:  1   4  10  20  35  56  84 120 165
  n=7:  1   2   3   4   5   6   7   8   9
  n=8:  1   4  10  20  35  56  84 120 165
  n=9:  1   3   6  10  15  21  28  36  45
Triangle begins:
   1
   1   1
   1   2   1
   1   3   2   1
   1   4   3   3   1
   1   5   4   6   2   1
   1   6   5  10   3   4   1
   1   7   6  15   4  10   2   1
   1   8   7  21   5  20   3   4   1
   1   9   8  28   6  35   4  10   3   1
   1  10   9  36   7  56   5  20   6   4   1
   1  11  10  45   8  84   6  35  10  10   2   1
For example, row n = 6 counts the following multisets:
  {1,1,1,1,1}  {1,1,1,1}  {1,1,1}  {1,1}  {1}  {}
               {1,1,1,2}  {1,1,3}  {1,2}  {5}
               {1,1,2,2}  {1,3,3}  {1,4}
               {1,2,2,2}  {3,3,3}  {2,2}
               {2,2,2,2}           {2,4}
                                   {4,4}
Note that for n = 6, k = 4 in the triangle, the two multisets {1,4} and {2,2} represent the same divisor 4, so they are only counted once under A343656(4,2) = 5.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

Row k = 1 of the array is A000005.
Column n = 4 of the array is A000217.
Column n = 6 of the array is A000292.
Row k = 2 of the array is A184389.
The distinct products of these multisets are counted by A343656.
Antidiagonal sums of the array (or row sums of the triangle) are A343661.
A000312 = n^n.
A009998(n,k) = n^k (as an array, offset 1).
A007318 counts k-sets of elements of {1..n}.
A059481 counts k-multisets of elements of {1..n}.
Cf. A000169, A062319, A067824, A143773, A146291, A176029, A285572, A326077, A327527, A334996, A343652, A343657.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,k],n-k],{n,10},{k,n}]

A(n,k) = binomial(numdiv(n) + k - 1, k)
{ for(n=1, 9, for(k=0, 8, print1(A(n,k), ", ")); print ) } \\ Andrew Howroyd, Jan 11 2024

A(n,k) = ((A000005(n), k)) = A007318(A000005(n) + k - 1, k).

T(n,k) = ((A000005(k), n - k)) = A007318(A000005(k) + n - k - 1, n - k).

A049017 Expansion of 1/((1-x)^7 - x^7).

Original entry on oeis.org

1, 7, 28, 84, 210, 462, 924, 1717, 3017, 5110, 8568, 14756, 27132, 54264, 116281, 257775, 572264, 1246784, 2641366, 5430530, 10861060, 21242341, 40927033, 78354346, 150402700, 291693136, 574274008, 1148548016, 2326683921, 4749439975, 9714753412, 19818498700, 40199107690

Offset: 0

N. J. A. Sloane

Differs for n >= 7 (1717 vs. 1716) from A000579(n+6) = binomial(n+6,6); see also row 6 of A027555, A059481 and A213808. - M. F. Hasler, Mar 05 2017

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,2).

Cf. A000579, A000750, A027555, A049018, A059481, A213808.

Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), A049016 (m=5), A192080 (m=6), this sequence (m=7), A290995 (m=8), A306939 (m=9).

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x)^7 - x^7) )); // G. C. Greubel, Apr 11 2023

CoefficientList[Series[1/((1-x)^7-x^7),{x,0,30}],x]  (* Harvey P. Dale, Feb 18 2011 *)

Vec(1/((1-x)^7-x^7)+O(x^99)) \\ M. F. Hasler, Mar 05 2017

def A049017_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)^7 - x^7) ).list()
A049017_list(40) # G. C. Greubel, Apr 11 2023

G.f.: 1/((1-x)^7 - x^7) = 1/((1-2*x)*(1-5*x+11*x^2-13*x^3+9*x^4-3*x^5+x^6)).

A123160 Triangle read by rows: T(n,k) = n!(n+k-1)!/((n-k)!(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 9, 18, 10, 1, 16, 60, 80, 35, 1, 25, 150, 350, 350, 126, 1, 36, 315, 1120, 1890, 1512, 462, 1, 49, 588, 2940, 7350, 9702, 6468, 1716, 1, 64, 1008, 6720, 23100, 44352, 48048, 27456, 6435, 1, 81, 1620, 13860, 62370, 162162, 252252, 231660, 115830, 24310

Offset: 0

Roger L. Bagula, Oct 02 2006

T(n,k) is also the number of order-preserving partial transformations (of an n-element chain) of width k (width(alpha) = |Dom(alpha)|). - Abdullahi Umar, Aug 25 2008

			Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1,  9,  18,  10;
  1, 16,  60,  80,  35;
  1, 25, 150, 350, 350, 126;
  ...

Frederick T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco, 1965 pages 296 and 305

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.

Cf. A005773, A037965, A059481, A085373, A088218, A088617, A122899, A123164, A178301.

[Binomial(n,k)*Binomial(n+k-1,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2022

T:=proc(n,k) if k=0 and n=0 then 1 elif k<=n then n!*(n+k-1)!/(n-k)!/(n-1)!/(k!)^2 else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

T[n_, m_]= If [n==m==0, 1, n!*(n+m-1)!/((n-m)!*(n-1)!(m!)^2)];
Table[T[n, m], {n,0,10}, {m,0,n}]//Flatten
max = 9; s = (x+1)/(2*Sqrt[(1-x)^2-4*y])+1/2 + O[x]^(max+2) + O[y]^(max+2); T[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]; Table[T[n-k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 18 2015, after Vladimir Kruchinin *)

def A123160(n,k): return binomial(n, k)*binomial(n+k-1, k)
flatten([[A123160(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2022

T(n, m) = n!*(n + m - 1)!/((n - m)!*(n - 1)!(m!)^2), with T(0, 0) = 1.
T(n, k) = binomial(n,k)*binomial(n+k-1,k). The row polynomials (except the first) are (1+x)*P(n,0,1,2x+1), where P(n,a,b,x) denotes the Jacobi polynomial. The columns of this triangle give the diagonals of A122899. - Peter Bala, Jan 24 2008
T(n, k) = binomial(n,k)*(n+k-1)!/((n-1)!*k!).
T(n, k)= binomial(n,k)*binomial(n+k-1,n-1). - Abdullahi Umar, Aug 25 2008
G.f.: (x+1)/(2*sqrt((1-x)^2-4*y)) + 1/2. - Vladimir Kruchinin, Jun 16 2015
From _Peter Bala, Jul 20 2015: (Start)
O.g.f. (1 + x)/( 2*sqrt((1 - x)^2 - 4*x*y) ) + 1/2 = 1 + (1 + y)*x + (1 + 4*y + 3*y^2)*x^2 + ....
For n >= 1, the n-th row polynomial R(n,y) = (1 + y)*r(n-1,y), where r(n,y) is the n-th row polynomial of A178301.
exp( Sum_{n >= 1} R(n,y)*x^n/n ) = 1 + (1 + y)*x + (1 + 3*y + 2*y^2)*x^2 + ... is the o.g.f for A088617. (End)
From G. C. Greubel, Jun 19 2022: (Start)
T(n, n) = A088218(n).
T(n, n-1) = A037965(n).
T(n, n-2) = A085373(n-2).
Sum_{k=0..n} T(n, k) = A123164(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A005773(n). (End)

Edited by N. J. A. Sloane, Oct 26 2006 and Jul 03 2008

A343657 Sum of number of divisors of x^y for each x >= 1, y >= 0, x + y = n.

Original entry on oeis.org

1, 2, 4, 7, 12, 18, 27, 39, 56, 77, 103, 134, 174, 223, 283, 356, 445, 547, 666, 802, 959, 1139, 1344, 1574, 1835, 2128, 2454, 2815, 3213, 3648, 4126, 4653, 5239, 5888, 6608, 7407, 8298, 9288, 10385, 11597, 12936, 14408, 16025, 17799, 19746, 21882, 24221

Offset: 1

Gus Wiseman, Apr 29 2021

nonn

			The a(7) = 27 divisors:
  1  32  81  64  25  6  1
     16  27  32  5   3
     8   9   16  1   2
     4   3   8       1
     2   1   4
     1       2
             1

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Antidiagonal row sums (row sums of the triangle) of A343656.
Dominated by A343661.
A000005(n) counts divisors of n.
A000312(n) = n^n.
A007318(n,k) counts k-sets of elements of {1..n}.
A009998(n,k) = n^k (as an array, offset 1).
A059481(n,k) counts k-multisets of elements of {1..n}.
A343658(n,k) counts k-multisets of divisors of n.
Cf. A000169, A000272, A002064, A002109, A048691, A062319, A066959, A143773, A146291, A176029, A251683, A282935, A326358, A327527, A334996.

Total/@Table[DivisorSigma[0,k^(n-k)],{n,30},{k,n}]

a(n) = Sum_{k=1..n} A000005(k^(n-k)).

A343935 Number of ways to choose a multiset of n divisors of n.

Original entry on oeis.org

1, 3, 4, 15, 6, 84, 8, 165, 55, 286, 12, 6188, 14, 680, 816, 4845, 18, 33649, 20, 53130, 2024, 2300, 24, 2629575, 351, 3654, 4060, 237336, 30, 10295472, 32, 435897, 7140, 7770, 8436, 177232627, 38, 10660, 11480, 62891499, 42, 85900584, 44, 1906884, 2118760

Offset: 1

Gus Wiseman, May 05 2021

nonn

			The a(1) = 1 through a(5) = 6 multisets:
  {1}  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}
       {1,2}  {1,1,3}  {1,1,1,2}  {1,1,1,1,5}
       {2,2}  {1,3,3}  {1,1,1,4}  {1,1,1,5,5}
              {3,3,3}  {1,1,2,2}  {1,1,5,5,5}
                       {1,1,2,4}  {1,5,5,5,5}
                       {1,1,4,4}  {5,5,5,5,5}
                       {1,2,2,2}
                       {1,2,2,4}
                       {1,2,4,4}
                       {1,4,4,4}
                       {2,2,2,2}
                       {2,2,2,4}
                       {2,2,4,4}
                       {2,4,4,4}
                       {4,4,4,4}

Diagonal n = k of A343658.
Choosing n divisors of n - 1 gives A343936.
The version for chains of divisors is A343939.
A000005 counts divisors.
A000312 = n^n.
A007318 counts k-sets of elements of {1..n}.
A009998 = n^k (as an array, offset 1).
A059481 counts k-multisets of elements of {1..n}.
A146291 counts divisors of n with k prime factors (with multiplicity).
A253249 counts nonempty chains of divisors of n.
Strict chains of divisors:
- A067824 counts strict chains of divisors starting with n.
- A074206 counts strict chains of divisors from n to 1.
- A251683 counts strict length k + 1 chains of divisors from n to 1.
- A334996 counts strict length-k chains of divisors from n to 1.
- A337255 counts strict length-k chains of divisors starting with n.
- A337256 counts strict chains of divisors of n.
- A343662 counts strict length-k chains of divisors.
Cf. A062319, A143773, A163767, A176029, A327527, A334997, A343661.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,n],n],{n,25}]

from math import comb
from sympy import divisor_count
def A343935(n): return comb(divisor_count(n)+n-1,n) # Chai Wah Wu, Jul 05 2024

a(n) = ((sigma(n), n)) = binomial(sigma(n) + n - 1, n) where sigma = A000005 and binomial = A007318.

A213745 Triangle of numbers C^(6)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 6 appearances allowed.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 10, 20, 35, 1, 5, 15, 35, 70, 126, 1, 6, 21, 56, 126, 252, 462, 1, 7, 28, 84, 210, 462, 924, 1709, 1, 8, 36, 120, 330, 792, 1716, 3424, 6371, 1, 9, 45, 165, 495, 1287, 3003, 6426, 12789, 23905, 1, 10

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jun 19 2012

For k<=5, the triangle coincides with triangle A213744.

We have over columns of the triangle: T(n,0)=1, T(n,1)=n, T(n,2)=A000217(n) for n>1, T(n,3)=A000292(n) for n>=3, T(n,4)=A000332(n) for n>=7, T(n,5)=A000389(n) for n>=9, T(n,6)=A000579(n) for n>=11, T(n,7)=A063267 for n>=5, T(n,8)=A063417 for n>=6, T(n,9)=A063418 for n>=7.

			Triangle begins
n/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1.....2.....3
.3..|..1.....3.....6....10
.4..|..1.....4....10....20....35
.5..|..1.....5....15....35....70....126
.6..|..1.....6....21....56...126....252...462
.7..|..1.....7....28....84...210....462...924....1709

Peter J. C. Moses, Rows n = 0..50 of triangle, flattened

Cf. A007318, A005725, A059481, A111808, A187925, A213742, A213743, A213744, A000217, A000292, A000332, A000389, A000579, A063267, A063417, A063418.

Flatten[Table[Sum[(-1)^r Binomial[n,r] Binomial[n-# r+k-1,n-1],{r,0,Floor[k/#]}],{n,0,15},{k,0,n}]/.{0}->{1}]&[7] (* Peter J. C. Moses, Apr 16 2013 *)

C^(6)(n,k)=sum{r=0,...,floor(k/7)}(-1)^r*C(n,r)*C(n-7*r+k-1, n-1).
A generalization. The numbers C^(t)(n,k) of combinations with repetitions from n different elements over k, for each of them not more than t>=1 appearances allowed, are enumerated by the formula:
C^(t)(n,k)=sum{r=0,...,floor(k/(t+1))}(-1)^r*C(n,r)*C(n-(t+1)*r+k-1, n-1).
In case t=1, it is binomial coefficient C^(t)(n,k)=C(n,k), and we have the combinatorial identity: sum{r=0,...,floor(k/2)}(-1)^r*C(n,r)*C(n-2*r+k-1, n-1)=C(n,k). On the other hand, if t=n, then r=0, and for the corresponding numbers of combinations with repetitions without a restriction on appearances of elements we obtain a well known formula C(n+k-1, n-1) (cf. triangle A059481).
In addition, note that, if k<=t, then C^(t)(n,k)=C(n+k-1, n-1). Therefore, triangle {C^(t+1)(n,k)} coincides with the previous triangle {C^(t)(n,k)} for k<=t.

A343661 Sum of numbers of y-multisets of divisors of x for each x >= 1, y >= 0, x + y = n.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 46, 70, 105, 155, 223, 316, 443, 619, 865, 1210, 1690, 2354, 3263, 4497, 6157, 8368, 11280, 15078, 19989, 26296, 34356, 44626, 57693, 74321, 95503, 122535, 157101, 201377, 258155, 330994, 424398, 544035, 696995, 892104, 1140298, 1455080

Offset: 1

Gus Wiseman, Apr 30 2021

nonn

			The a(5) = 12 multisets of divisors:
  {1,1,1,1}  {1,1,1}  {1,1}  {1}  {}
             {1,1,2}  {1,3}  {2}
             {1,2,2}  {3,3}  {4}
             {2,2,2}

Antidiagonal sums of the array A343658 (or row sums of the triangle).
Dominates A343657.
A000005 counts divisors.
A007318 counts k-sets of elements of {1..n}.
A059481 counts k-multisets of elements of {1..n}.
A343656 counts divisors of powers.
Cf. A000169, A000312, A009998, A062319, A067824, A143773, A146291, A176029, A184389, A285572, A326077, A327527, A334996, A343652, A343657.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[Sum[multchoo[DivisorSigma[0,k],n-k],{k,n}],{n,10}]

a(n) = Sum_{k=1..n} binomial(sigma(k) + n - k - 1, n - k).

A165257 Triangle in which n-th row is binomial(n+k-1,k), for column k=1..n.

Original entry on oeis.org

1, 2, 3, 3, 6, 10, 4, 10, 20, 35, 5, 15, 35, 70, 126, 6, 21, 56, 126, 252, 462, 7, 28, 84, 210, 462, 924, 1716, 8, 36, 120, 330, 792, 1716, 3432, 6435, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310

Offset: 1

Daniel McLaury (daniel.mclaury(AT)gmail.com), Sep 11 2009

T(n,k) is the number of non-descending sequences with length k and last number is less or equal to n. T(n,k) is also the number of integer partitions (of any positive integer) with length k and largest part is less or equal to n. - Zlatko Damijanic, Dec 06 2024

			1;
2, 3;
3, 6, 10;
4, 10, 20, 35;
5, 15, 35, 70, 126;
6, 21, 56, 126, 252, 462;
7, 28, 84, 210, 462, 924, 1716;

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A059481 with the first column (k = 0) removed.

Cf. A030662 (row sums), A001700 (diagonal).

Table[Binomial[n+k-1,k],{n,10},{k,n}]//Flatten (* Harvey P. Dale, Jul 31 2021 *)

tabl(nn) = {for (n=1, nn, for (k=1, n, print1( binomial(n+k-1, k), ", ");); print(););} \\ Michel Marcus, Jun 12 2013

A328745 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s))^p.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 4, 6, 10, 11, 9, 13, 14, 15, 5, 17, 12, 19, 15, 21, 22, 23, 12, 15, 26, 10, 21, 29, 30, 31, 6, 33, 34, 35, 18, 37, 38, 39, 20, 41, 42, 43, 33, 30, 46, 47, 15, 28, 30, 51, 39, 53, 20, 55, 28, 57, 58, 59, 45, 61, 62, 42, 7, 65, 66, 67, 51, 69, 70, 71, 24, 73, 74, 45

Offset: 1

Ilya Gutkovskiy, Oct 26 2019

Number of ways to factor n into 2 kinds of 2, 3 kinds of 3, 5 kinds of 5, ... , p kinds of p.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000

Cf. A008480, A050367, A059481, A135323, A182938.

a:= n-> mul(binomial(i[1]+i[2]-1, i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 26 2019

a[n_] := Times @@ (Binomial[#[[1]] + #[[2]] - 1, #[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 75}]

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X)^p)[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021

If n = Product (p_j^k_j) then a(n) = Product (binomial(p_j + k_j - 1, k_j)).

Conjecture: Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.40373... - Vaclav Kotesovec, Mar 28 2025

cp's OEIS Frontend

A027555 Triangle of binomial coefficients C(-n,k).

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A343658 Array read by antidiagonals where A(n,k) is the number of ways to choose a multiset of k divisors of n.

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A049017 Expansion of 1/((1-x)^7 - x^7).

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A123160 Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.

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A343657 Sum of number of divisors of x^y for each x >= 1, y >= 0, x + y = n.

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A343935 Number of ways to choose a multiset of n divisors of n.

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A213745 Triangle of numbers C^(6)(n,k) of combinations with repetitions from n different elements over k for each of them not more than 6 appearances allowed.

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A123160 Triangle read by rows: T(n,k) = n!(n+k-1)!/((n-k)!(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.