A101508 -id:A101508 - cp's OEIS frontend

A051731 Triangle read by rows: T(n, k) = 1 if k divides n, T(n, k) = 0 otherwise, for 1 <= k <= n.

1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

T(n, k) is the number of partitions of n into k equal parts. - Omar E. Pol, Apr 21 2018

This triangle is the lower triangular array L in the LU decomposition of the square array A003989. - Peter Bala, Oct 15 2023

			The triangle T(n, k) begins:
  n\k 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
  1:  1
  2:  1  1
  3:  1  0  1
  4:  1  1  0  1
  5:  1  0  0  0  1
  6:  1  1  1  0  0  1
  7:  1  0  0  0  0  0  1
  8:  1  1  0  1  0  0  0  1
  9:  1  0  1  0  0  0  0  0  1
  10: 1  1  0  0  1  0  0  0  0  1
  11: 1  0  0  0  0  0  0  0  0  0  1
  12: 1  1  1  1  0  1  0  0  0  0  0  1
  13: 1  0  0  0  0  0  0  0  0  0  0  0  1
  14: 1  1  0  0  0  0  1  0  0  0  0  0  0  1
  15: 1  0  1  0  1  0  0  0  0  0  0  0  0  0  1
  ... Reformatted and extended. - _Wolfdieter Lang_, Nov 12 2014

Charles R Greathouse IV, Rows n = 1..100, flattened
Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.
Mats Granvik, Illustration.
Warren P. Johnson, An LDU Factorization in Elementary Number Theory, Mathematics Magazine, Vol. 76, No. 5 (Dec., 2003), pp. 392-394.
Jeffrey Ventrella, Divisor Plot.

Cf. A000007, A000203, A002260, A003989, A048158, A049820, A054525.
Cf. A074854, A101508, A115361, A129372, A172119, A175105.
Cf. A000005 (row sums), A032741(n+2) (diagonal sums).
Cf. A243987 (partial sums per row).
Cf. A134546 (A004736 * T, matrix multiplication).
Variants: A113704, A077049, A077051.

a051731 n k = 0 ^ mod n k
a051731_row n = a051731_tabl !! (n-1)
a051731_tabl = map (map a000007) a048158_tabl
-- Reinhard Zumkeller, Aug 13 2013

[0^(n mod k): k in [1..n], n in [1..17]]; // G. C. Greubel, Jun 22 2024

A051731 := proc(n, k) if n mod k = 0 then 1 else 0 end if end proc:
# R. J. Mathar, Jul 14 2012

Flatten[Table[If[Mod[n, k] == 0, 1, 0], {n, 20}, {k, n}]]

for(n=1,17,for(k=1,n,print1(!(n%k)", "))) \\ Charles R Greathouse IV, Mar 14 2012

from math import isqrt, comb
def A051731(n): return int(not (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))%(n-comb(a,2))) # Chai Wah Wu, Nov 13 2024

A051731_row = lambda n: [int(k.divides(n)) for k in (1..n)]
for n in (1..17): print(A051731_row(n)) # Peter Luschny, Jan 05 2018

{T(n, k)*k, k=1..n} setminus {0} = divisors(n).
Sum_{k=1..n} T(n, k)*k^i = sigma[i](n), where sigma[i](n) is the sum of the i-th power of the positive divisors of n.
Sum_{k=1..n} T(n, k) = A000005(n).
Sum_{k=1..n} T(n, k)*k = A000203(n).
T(n, k) = T(n-k, k) for k <= n/2, T(n, k) = 0 for n/2 < k <= n-1, T(n, n) = 1.
Rows given by A074854 converted to binary. Example: A074854(4) = 13 = 1101_2; row 4 = 1, 1, 0, 1. - Philippe Deléham, Oct 04 2003
From Paul Barry, Dec 05 2004: (Start)
Binomial transform (product by binomial matrix) is A101508.
Columns have g.f.: x^k/(1-x^(k+1)) (k >= 0). (End)
Matrix inverse of triangle A054525, where A054525(n, k) = MoebiusMu(n/k) if k|n, 0 otherwise. - Paul D. Hanna, Jan 09 2006
From Gary W. Adamson, Apr 15 2007, May 10 2007: (Start)
Equals A129372 * A115361 as infinite lower triangular matrices.
A054525 is the inverse of this triangle (as lower triangular matrix).
This triangle * [1, 2, 3, ...] = sigma(n) (A000203).
This triangle * [1/1, 1/2, 1/3, ...] = sigma(n)/n. (End)
From  Reinhard Zumkeller, Nov 01 2009: (Start)
T(n, k) = 0^(n mod k).
T(n, k) = A000007(A048158(n, k)). (End)
From Mats Granvik, Jan 26 2010, Feb 10 2010, Feb 16 2010: (Start)
T(n, k) = A172119(n) mod 2.
T(n, k) = A175105(n) mod 2.
T(n, k) = Sum_{i=1..k-1} (T(n-i, k-1) - T(n-i, k)) for k > 1 and T(n, 1) = 1.
(Jeffrey O. Shallit kindly provided a clarification along with a proof of this formula.) (End)
A049820(n) = number of zeros in n-th row. - Reinhard Zumkeller, Mar 09 2010
The determinant of this matrix where T(n, n) has been swapped with T(1,k) is equal to the n-th term of the Mobius function. - Mats Granvik, Jul 21 2012
T(n, k) = Sum_{y=1..n} Sum_{x=1..n} [GCD((x/y)*(k/n), n) = k]. - Mats Granvik, Dec 17 2023

Edited by Peter Luschny, Oct 18 2023

A101509 Binomial transform of tau(n) (see A000005).

Original entry on oeis.org

1, 3, 7, 16, 35, 75, 159, 334, 696, 1442, 2976, 6123, 12562, 25706, 52492, 107014, 217877, 443061, 899957, 1826078, 3701783, 7498261, 15178255, 30706320, 62085915, 125465715, 253415981, 511608490, 1032427637, 2082680887, 4199956101, 8467124805, 17064784905, 34382825363, 69256687719, 139465867773

Offset: 0

Paul Barry, Dec 05 2004

Row sums of A101508.

Also: Number of matrices with positive integer coefficients such that the sum of all entries equals n+1, cf. link "Partitions and A101509". - M. F. Hasler, Jan 14 2009

			From _Gus Wiseman_, Jan 16 2019: (Start)
The a(3) = 16 ways to arrange the parts of an integer partition of 4 into a matrix:
  [4] [1 3] [3 1] [2 2] [1 1 2] [1 2 1] [2 1 1] [1 1 1 1]
.
  [1] [3] [2] [1 1]
  [3] [1] [2] [1 1]
.
  [1] [1] [2]
  [1] [2] [1]
  [2] [1] [1]
.
  [1]
  [1]
  [1]
  [1]
(End)

M. F. Hasler, Table of n, a(n) for n = 0..500
L. Manor, M. F. Hasler, Partitions and A101509. SeqFan list, Jan 14 2009
N. J. A. Sloane, Transforms

Cf. A000005 (tau), A101508, A160399.

Cf. A000219, A047966, A053529, A319066, A323300, A323301, A323307, A323429.

bintr:= proc(p) proc(n) add(p(k) *binomial(n, k), k=0..n) end end:
a:= bintr(n-> numtheory[tau](n+1)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 30 2011

a[n_] := Sum[DivisorSigma[0, k+1]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2017 *)

A101509(n) = sum( k=0,n, numdiv(k+1)*binomial(n,k)) \\ M. F. Hasler, Jan 14 2009

a(n) = Sum_{k=0..n, Sum_{i=0..n, if(mod(i+1, k+1)=0, binomial(n, i), 0)}}.
G.f.: 1/x * Sum_{n>=1} z^n/(1-z^n) (Lambert series) where z=x/(1-x). - Joerg Arndt, Jan 30 2011
a(n) ~ 2^n * (log(n/2) + 2*gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 07 2020

A306915 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1-x)^k-x^k).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 3, 4, 8, 1, 4, 6, 8, 16, 1, 5, 10, 11, 16, 32, 1, 6, 15, 20, 21, 32, 64, 1, 7, 21, 35, 36, 42, 64, 128, 1, 8, 28, 56, 70, 64, 85, 128, 256, 1, 9, 36, 84, 126, 127, 120, 171, 256, 512, 1, 10, 45, 120, 210, 252, 220, 240, 342, 512, 1024

Offset: 0

Seiichi Manyama, Mar 16 2019

			Square array begins:
     1,   1,   1,   1,   1,    1,    1,    1, ...
     2,   2,   3,   4,   5,    6,    7,    8, ...
     4,   4,   6,  10,  15,   21,   28,   36, ...
     8,   8,  11,  20,  35,   56,   84,  120, ...
    16,  16,  21,  36,  70,  126,  210,  330, ...
    32,  32,  42,  64, 127,  252,  462,  792, ...
    64,  64,  85, 120, 220,  463,  924, 1716, ...
   128, 128, 171, 240, 385,  804, 1717, 3432, ...
   256, 256, 342, 496, 715, 1365, 3017, 6436, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns (1+2),3-9 give A000079, A024495(n+2), A000749(n+3), A049016, A192080, A049017, A290995(n+7), A306939.

Cf. A039912, A101508, A306846, A306913, A306914, A307047, A307078, A307393.

A[n_, k_] := Sum[Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[A[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 25 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+k-1,k*j+k-1).

A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i+k-1,k*j+k-1) * binomial(n-i+k-1,k*j+k-1). - Seiichi Manyama, Apr 07 2019

A101510 Diagonal sums of binomial-Möbius product.

Original entry on oeis.org

1, 2, 5, 10, 21, 43, 87, 175, 352, 707, 1417, 2836, 5674, 11353, 22716, 45443, 90886, 181748, 363451, 726870, 1453773, 2907648, 5815315, 11630195, 23259059, 46515887, 93029852, 186060921, 372129424, 744272221, 1488552317, 2977079872

Offset: 0

Paul Barry, Dec 05 2004

Diagonal sums of A101508.

Seiichi Manyama, Table of n, a(n) for n = 0..500

a[n_] := Sum[If[Mod[i+1, k+1] == 0, Binomial[n-k, i], 0], {k, 0, n/2}, {i, 0, n-k}]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jan 24 2014 *)
nmax = 40; CoefficientList[Series[(1/x^2) * Sum[x*(x/(1-x))^k/(1-x*(x/(1-x))^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 18 2019 *)

a(n) = sum(k=0, n\2, sum(i=0, n-k, if (!Mod(i+1, k+1), binomial(n-k, i)))); \\ Michel Marcus, Mar 16 2019

a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..n-k, (k+1)|(i+1)} binomial(n-k,i).
G.f.: (1/x^2) * Sum_{n>=1} a*z^n/(1-a*z^n) (generalized Lambert series) where z=x/(1-x) and a=x. - Joerg Arndt, Jan 30 2011
a(n) ~ log(2) * 2^(n+1). - Vaclav Kotesovec, Mar 18 2019

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A051731 Triangle read by rows: T(n, k) = 1 if k divides n, T(n, k) = 0 otherwise, for 1 <= k <= n.

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A101509 Binomial transform of tau(n) (see A000005).

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A306915 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1-x)^k-x^k).

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A101510 Diagonal sums of binomial-Möbius product.

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