A109974 -id:A109974 - cp's OEIS frontend

A109976 Diagonal sums of number triangle A109974.

1, 2, 3, 6, 7, 16, 19, 46, 66, 159, 274, 667, 1320, 3263, 7156, 18084, 42794, 111018, 278752, 743889, 1959482, 5383395, 14761634, 41740307, 118517301, 344580865, 1009349749, 3013665637, 9081346743, 27808754039, 86015003997, 269796018547

Offset: 0

Paul Barry, Jul 06 2005

Robert Israel, Table of n, a(n) for n = 0..1063

Cf. A109974

f:= n -> add(numtheory:-sigma[k](n-2*k+1),k=0..n/2):
map(f, [$0..50]); # Robert Israel, May 25 2018

Array[Sum[DivisorSigma[k, (# - 2 k + 1)], {k, 0, Floor[#/2]}] &, 32, 0] (* Michael De Vlieger, May 27 2018 *)

a(n) = Sum_{k=0..floor(n/2)} sigma_k(n-2k+1). - Corrected by Robert Israel, May 25 2018

G.f.: Sum_{n>=1} x^(n-1)/((1-x^n)*(1-n*x^2)). - Robert Israel, May 27 2018

A109977 Inverse of number-theoretic triangle A109974.

Original entry on oeis.org

1, -2, 1, 4, -3, 1, -15, 11, -5, 1, 107, -76, 35, -9, 1, -1475, 1041, -476, 125, -17, 1, 40914, -28858, 13177, -3460, 479, -33, 1, -2327019, 1641270, -749336, 196731, -27260, 1901, -65, 1, 271813931, -191712888, 87527566, -22979242, 3184213, -222196, 7655, -129, 1, -64930439315, 45796039830

Offset: 0

Paul Barry, Jul 06 2005

			Rows begin
1;
-2,1;
4,-3,1;
-15,11,-5,1;
107,-76,35,-9,1;
-1475,1041,-476,125,-17,1;

A109978 Inverse binomial transform of number-theoretic triangle A109974.

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 2, -2, 2, 1, -5, 5, -4, 5, 1, 13, -14, 11, -7, 12, 1, -33, 40, -35, 20, -5, 27, 1, 80, -111, 113, -77, 21, 34, 58, 1, -184, 293, -350, 294, -144, -27, 238, 121, 1, 402, -731, 1021, -1042, 716, -249, -153, 1063, 248, 1, -840, 1726, -2796, 3409, -2982, 1755, -724, 318, 4037, 503, 1

Offset: 1

Paul Barry, Jul 06 2005

First column is expansion of bracket function A001659.

			Rows begin
1;
1,1;
-1,1,1;
2,-2,2,1;
-5,5,-4,5,1;
13,-14,11,-7,12,1;
-33,40,-35,20,-5,27,1;

T(n, k)=sum{j=1..n, (-1)^(n-j)C(n-1, j-1)*if(k<=j, sigma(k-1, j-k+1), 0)} [offset (1, 1)]

A023887 a(n) = sigma_n(n): sum of n-th powers of divisors of n.

Original entry on oeis.org

1, 5, 28, 273, 3126, 47450, 823544, 16843009, 387440173, 10009766650, 285311670612, 8918294543346, 302875106592254, 11112685048647250, 437893920912786408, 18447025552981295105, 827240261886336764178, 39346558271492178925595, 1978419655660313589123980

Offset: 1

Olivier Gérard

Logarithmic derivative of A023881.

Compare to A217872(n) = sigma(n)^n.

			The divisors of 6 are 1, 2, 3 and 6, so a(6) = 1^6 + 2^6 + 3^6 + 6^6 = 47450.

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

Nick Hobson, Table of n, a(n) for n = 1..100

Cf. A000203, A001157-A001160, A013954-A013972, A023881, A199858, subdiagonal in A109974.

A023887 := proc(n)
    numtheory[sigma][n](n) ;
end proc:
seq(A023887(n),n=1..10) ; # R. J. Mathar, Apr 06 2022

Table[DivisorSigma[n,n],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)

makelist(divsum(n,n),n,1,20); /* Emanuele Munarini, Mar 26 2011 */

a(n) = sigma(n,n); \\ Nick Hobson, Nov 25 2006

from sympy import divisor_sigma
def A023887(n): return divisor_sigma(n,n) # Chai Wah Wu, Jun 19 2022

G.f.: Sum_{n>0} (n*x)^n/(1-(n*x)^n). - Vladeta Jovovic, Oct 27 2002
From Nick Hobson, Nov 25 2006: (Start)
If the canonical prime factorization of n > 1 is the product of p^e(p) then sigma_n(n) = Product_p ((p^(n*(e(p)+1)))-1)/(p^n-1).
sigma_n(n) is odd if and only if n is a square or twice a square. (End)
Conjecture: sigma_m(n) = sigma(n^m * rad(n)^(m-1))/sigma(rad(n)^(m-1)) for n > 0 and m > 0, where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 24 2017
a(n) ~ n^n. - Vaclav Kotesovec, Nov 02 2018
Sum_{n>=1} 1/a(n) = A199858. - Amiram Eldar, Nov 19 2020

Edited by N. J. A. Sloane, Nov 25 2006

A322083 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+d)*d^k.

Original entry on oeis.org

1, 1, -2, 1, -3, 2, 1, -5, 4, -1, 1, -9, 10, -3, 2, 1, -17, 28, -13, 6, -4, 1, -33, 82, -57, 26, -12, 2, 1, -65, 244, -241, 126, -50, 8, 0, 1, -129, 730, -993, 626, -252, 50, -3, 3, 1, -257, 2188, -4033, 3126, -1394, 344, -45, 13, -4, 1, -513, 6562, -16257, 15626, -8052, 2402, -441, 91, -18, 2

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

For each k, the k-th column sequence (T(n,k))(n>=1) is a multiplicative function of n, equal to (-1)^(n+1)*(Id_k * 1) in the notation of the Bala link. - Peter Bala, Mar 19 2022

			Square array begins:
   1,   1,   1,    1,     1,     1,  ...
  -2,  -3,  -5,   -9,   -17,   -33,  ...
   2,   4,  10,   28,    82,   244,  ...
  -1,  -3, -13,  -57,  -241,  -993,  ...
   2,   6,  26,  126,   626,  3126,  ...
  -4, -12, -50, -252, -1394, -8052,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Peter Bala, A signed Dirichlet product of arithmetical functions
Index entries for sequences mentioned by Glaisher

Columns k=0..12 give A228441, A109506, A321558, A321559, A321560, A321561, A321562, A321563, A321564, A321565, A321807, A321808, A321809.

Cf. A109974, A279394, A279396, A285425, A322081, A322082, A322084.

Table[Function[k, Sum[(-1)^(n/d+d) d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[(-1)^(j + 1) j^k x^j/(1 + x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
f[p_, e_, k_] := If[k == 0, e + 1, (p^(k*e + k) - 1)/(p^k - 1)]; f[2, e_, k_] := If[k == 0, e - 3, -((2^(k - 1) - 1)*2^(k*e + 1) + 2^(k + 1) - 1)/(2^k - 1)]; T[1, k_] = 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[n - k, k], {n, 1, 11}, {k, n - 1, 0, -1}] // Flatten (* Amiram Eldar, Nov 22 2022 *)

T(n,k)={sumdiv(n, d, (-1)^(n/d+d)*d^k)}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

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

A285425 Square array A(n,k), n>=1, k>=0, read by antidiagonals, where column k is the expansion of Sum_{j>=1} (2j - 1)^kx^(2j-1)/(1 - x^(2j-1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 10, 1, 2, 1, 1, 28, 1, 6, 2, 1, 1, 82, 1, 26, 4, 2, 1, 1, 244, 1, 126, 10, 8, 1, 1, 1, 730, 1, 626, 28, 50, 1, 3, 1, 1, 2188, 1, 3126, 82, 344, 1, 13, 2, 1, 1, 6562, 1, 15626, 244, 2402, 1, 91, 6, 2, 1, 1, 19684, 1, 78126, 730, 16808, 1, 757, 26, 12, 2

Offset: 1

Ilya Gutkovskiy, May 14 2017

A(n,k) is the sum of k-th powers of odd divisors of n.

			Square array begins:
1,  1,   1,    1,    1,     1,  ...
1,  1,   1,    1,    1,     1,  ...
2,  4,  10,   28,   82,   244,  ...
1,  1,   1,    1,    1,     1,  ...
2,  6,  26,  126,  626,  3126,  ...
2,  4,  10,   28,   82,   244,  ...

Eric Weisstein's World of Mathematics, Odd Divisor Function
Index entries for sequences related to sums of divisors

Columns k=0-5 give: A001227, A000593, A050999, A051000, A051001, A051002.

Cf. A109974, A279394, A292919.

Table[Function[k, SeriesCoefficient[Sum[(2 i - 1)^k x^(2 i - 1)/(1 - x^(2 i - 1)), {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 1, j}] // Flatten

G.f. of column k: Sum_{j>=1} (2*j - 1)^k*x^(2*j-1)/(1 - x^(2*j-1)).

Offset changed by Ilya Gutkovskiy, Oct 25 2018

A279394 Triangle read by rows, T(n,m) = sigma_{n-m}(m) for n >= 1, m = 1,2, ..., n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 9, 10, 7, 2, 1, 17, 28, 21, 6, 4, 1, 33, 82, 73, 26, 12, 2, 1, 65, 244, 273, 126, 50, 8, 4, 1, 129, 730, 1057, 626, 252, 50, 15, 3, 1, 257, 2188, 4161, 3126, 1394, 344, 85, 13, 4, 1, 513, 6562, 16513, 15626, 8052, 2402, 585, 91, 18, 2, 1, 1025, 19684, 65793, 78126, 47450, 16808, 4369, 757, 130, 12, 6

Offset: 1

Wolfdieter Lang, Jan 07 2017

See A109974 (downward antidiagonals) for details and references. sigma_k(n) is the sum of the k-th power of the positive divisors of n.
This is the triangle read by rows obtained from the array sigma_k(n) for k >= 0, n >= 1, read by upward antidiagonals.
The row sums are A108639.

			The triangle T(n, m) begins:
n\m 1   2    3    4    5    6   7  8  9 10
1:  1
2:  1   2
3:  1   3    2
4:  1   5    4    3
5:  1   9   10    7    2
6:  1  17   28   21    6    4
7:  1  33   82   73   26   12   2
8:  1  65  244  273  126   50   8  4
9:  1 129  730 1057  626  252  50 15  3
10: 1 257 2188 4161 3126 1394 344 85 13  4
...
n = 11: 1 513 6562 16513 15626 8052 2402 585 91 18 2,
n = 12: 1 1025 19684 65793 78126 47450 16808 4369 757 130 12 6.
...

Cf. A109974, A108639.

T := (n, k) -> numtheory:-sigma[n-k](k):
seq(seq(T(n,k), k=1..n), n=1..12); # Peter Luschny, Jan 07 2017

Table[DivisorSigma[k, #] &[n - k + 1], {n, 0, 11}, {k, n, 0, -1}] (* Michael De Vlieger, Jan 09 2017 *)

T(n, m) = sigma_{n-m}(m), n >= 1, m = 1..n.

A225816 Square array A(n,k) = (k!)^n, n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 4, 1, 1, 1, 24, 36, 8, 1, 1, 1, 120, 576, 216, 16, 1, 1, 1, 720, 14400, 13824, 1296, 32, 1, 1, 1, 5040, 518400, 1728000, 331776, 7776, 64, 1, 1, 1, 40320, 25401600, 373248000, 207360000, 7962624, 46656, 128, 1, 1

Offset: 0

Alois P. Heinz, Jul 29 2013

A(n,k) is the determinant of the k X k matrix M = [Stirling2(n+i,j)] for 1<=i,j<=k.  A(2,3) = det([1,3,1; 1,7,6; 1,15,25]) = 36.
A(n,k) is the determinant of the symmetric k X k matrix M = [sigma_n(gcd(i,j))] for 1<=i,j<=k.  A(2,3) = det([1,1,1; 1,5,1; 1,1,10]) = 36.
A(n,k) is (-1)^(n*k) times the determinant of the n X n matrix M = [Stirling1(k+i,j)] for 1<=i,j<=n.  A(2,3) = (-1)^(2+3) * det([-6,11; 24,-50]) = 36.
A(n,k) is the number of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_k) we have abs(p_i-p_j) <= 1 for 1<=i,j<=k.  A(2,3) = 36:
          (1,2,2)-(1,1,2)         (0,1,1)-(0,0,1)
         /       X       \       /       X       \
  (2,2,2)-(2,1,2) (1,2,1)-(1,1,1)-(1,0,1) (0,1,0)-(0,0,0).
         \       X       /       \       X       /
          (2,2,1) (2,1,1)         (1,1,0) (1,0,0)
A(n,k) is the number of set partitions of [k*(n+1)] into k blocks of size n+1 such that the elements of each block are distinct mod n+1.  A(2,3) = 36: 123|456|789, 126|345|789, ..., 189|234|567, 189|246|357.

			Square array A(n,k) begins:
  1, 1,  1,    1,       1,           1, ...
  1, 1,  2,    6,      24,         120, ...
  1, 1,  4,   36,     576,       14400, ...
  1, 1,  8,  216,   13824,     1728000, ...
  1, 1, 16, 1296,  331776,   207360000, ...
  1, 1, 32, 7776, 7962624, 24883200000, ...

Alois P. Heinz, Antidiagonals n = 0..36, flattened

Columns k=0+1, 2-4 give: A000012, A000079, A000400, A009968.
Rows n=0-4 give: A000012, A000142, A001044, A000442, A134375.
Main diagonal gives: A036740.
Cf. A008275, A048994, A008277, A048993, A003989, A109004, A109974.

A:= (n, k)-> k!^n:
seq(seq(A(n,d-n), n=0..d), d=0..12);

A(n,k) = (k!)^n.
A(n,k) = k^n * A(n,k-1) for k>0, A(n,0) = 1.
A(n,k) = k!  * A(n-1,k) for n>0, A(0,k) = 1.
G.f. of column k: 1/(1-k!*x).

A322143 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, d==1 (mod 4)} d^k - Sum_{d|n, d==3 (mod 4)} d^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, -2, 1, 1, 1, -8, 1, 2, 1, 1, -26, 1, 6, 0, 1, 1, -80, 1, 26, -2, 0, 1, 1, -242, 1, 126, -8, -6, 1, 1, 1, -728, 1, 626, -26, -48, 1, 1, 1, 1, -2186, 1, 3126, -80, -342, 1, 7, 2, 1, 1, -6560, 1, 15626, -242, -2400, 1, 73, 6, 0, 1, 1, -19682, 1, 78126, -728, -16806, 1, 703, 26, -10, 0

Offset: 1

Ilya Gutkovskiy, Nov 28 2018

			Square array begins:
  1,  1,   1,    1,    1,     1,  ...
  1,  1,   1,    1,    1,     1,  ...
  0, -2,  -8,  -26,  -80,  -242,  ...
  1,  1,   1,    1,    1,     1,  ...
  2,  6,  26,  126,  626,  3126,  ...
  0, -2,  -8,  -26,  -80,  -242,  ...

Index entries for sequences mentioned by Glaisher

Columns k=0..12 give A002654, A050457, A002173, A050459, A050456, A321821, A321822, A321823, A321824, A321825, A321826, A321827, A321828.

Cf. A109974, A322081, A322082, A322083, A322084.

Table[Function[k, SeriesCoefficient[Sum[(-1)^(j - 1) (2 j - 1)^k x^(2 j - 1)/(1 - x^(2 j - 1)), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

G.f. of column k: Sum_{j>=1} (-1)^(j-1)*(2*j - 1)^k*x^(2*j-1)/(1 - x^(2*j-1)).

A286880 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where row n is the sum of n-th powers of unitary divisors of k (divisors d such that gcd(d, k/d) = 1).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 5, 10, 9, 1, 4, 6, 17, 28, 17, 1, 2, 12, 26, 65, 82, 33, 1, 2, 8, 50, 126, 257, 244, 65, 1, 2, 9, 50, 252, 626, 1025, 730, 129, 1, 4, 10, 65, 344, 1394, 3126, 4097, 2188, 257, 1, 2, 18, 82, 513, 2402, 8052, 15626, 16385, 6562, 513, 1, 4, 12, 130, 730, 4097, 16808, 47450, 78126, 65537, 19684, 1025, 1

Offset: 0

Ilya Gutkovskiy, Aug 02 2017

For row r > 0, Sum_{k=1..n} A(r,k) ~ zeta(r+1) * n^(r+1) / ((r+1) * zeta(r+2)). - Vaclav Kotesovec, May 20 2021

			Square array begins:
1,   2,    2,     2,     2,     4,  ...
1,   3,    4,     5,     6,    12,  ...
1,   5,   10,    17,    26,    50,  ...
1,   9,   28,    65,   126,   252,  ...
1,  17,   82,   257,   626,  1394,  ...
1,  33,  244,  1025,  3126,  8052,  ...

Eric Weisstein's World of Mathematics, Unitary Divisor
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Index entries for sequences related to sums of divisors

Rows n=0-8 give: A034444, A034448, A034676, A034677, A034678, A034679, A034680, A034681, A034682.

Cf. A077610, A109974, A285425.

Dirichlet g.f. of row n: zeta(s)*zeta(s-n)/zeta(2*s-n).

cp's OEIS Frontend

A109976 Diagonal sums of number triangle A109974.

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A109977 Inverse of number-theoretic triangle A109974.

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A109978 Inverse binomial transform of number-theoretic triangle A109974.

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A023887 a(n) = sigma_n(n): sum of n-th powers of divisors of n.

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A322083 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+d)*d^k.

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A285425 Square array A(n,k), n>=1, k>=0, read by antidiagonals, where column k is the expansion of Sum_{j>=1} (2*j - 1)^k*x^(2*j-1)/(1 - x^(2*j-1)).

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A279394 Triangle read by rows, T(n,m) = sigma_{n-m}(m) for n >= 1, m = 1,2, ..., n.

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A225816 Square array A(n,k) = (k!)^n, n>=0, k>=0, read by antidiagonals.

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A285425 Square array A(n,k), n>=1, k>=0, read by antidiagonals, where column k is the expansion of Sum_{j>=1} (2j - 1)^kx^(2j-1)/(1 - x^(2j-1)).