A279395 -id:A279395 - cp's OEIS frontend

A279396 Triangle read by rows T(n, m) = sigma^(n-m)(m), n >= 1, m = 1, 2, ..., n, with sigma^(k)(n) given in a comment in A279395.

1, 1, 0, 1, 1, 2, 1, 3, 4, 1, 1, 7, 10, 5, 2, 1, 15, 28, 19, 6, 0, 1, 31, 82, 71, 26, 4, 2, 1, 63, 244, 271, 126, 30, 8, 2, 1, 127, 730, 1055, 626, 196, 50, 13, 3, 1, 255, 2188, 4159, 3126, 1230, 344, 83, 13, 0, 1, 511, 6562, 16511, 15626, 7564, 2402, 583, 91, 6, 2, 1, 1023, 19684, 65791, 78126, 45990, 16808, 4367, 757, 78, 12, 2

Offset: 1

Wolfdieter Lang, Jan 10 2017

The array A(k, n) = sigma^*A279395)%20=%20Sum">(k)(n) (notation of the Hardy reference, given also in a comment in A279395) = Sum{ d >= 1, d divides n} (-1)^(n-d)*d^k, for k >= 0 and n >=1, has the rows A112329, A113184, A064027, A008457, A279395, for k=0..4.
The triangle T(n, m) is obtained from the array A(k, n) read by upwards antidiagonals, with offset n=1.
The diagonals of triangle T are the rows of the array A. Each diagonal is multiplicative. See the given A-numbers above.
The row sums are given in A279397.
The column sums (with offset 0) are A000012, A000225, A034472, A099393, A034474, .. with o.g.f. G(m, z) = (-1)^m*Sum_{d  | m} (-1)^d/(1 - d*z), m >= 1.

			The triangle T(n, m) begins:
n\m 1   2    3    4    5    6   7  8  9 10
1:  1
2:  1   0
3:  1   1    2
4:  1   3    4    1
5:  1   7   10    5    2
6:  1  15   28   19    6    0
7:  1  31   82   71   26    4   2
8:  1  63  244  271  126   30   8  2
9:  1 127  730 1055  626  196  50 13  3
10: 1 255 2188 4159 3126 1230 344 83 13  0
...
n = 11: 1 511 6562 16511 15626 7564 2402 583 91 6 2,
n = 12: 1 1023 19684 65791 78126 45990 16808 4367 757 78 12 2.
n = 13: 1 2047 59050 262655 390626 277876 117650 33823 6643 882 122 20 2,
n = 14: 1 4095 177148 1049599 1953126 1673310 823544 266303 59293 9390 1332 190 14 0,
n = 15: 1 8191 531442 4196351 9765626 10058524 5764802 2113663 532171 96906 14642 1988 170 8 4.
...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Cf. A279394, A112329, A113184, A064027, A008457, A279395, A000012, A000225, A034472, A099393, A034474.

T(n, m) = Sum_{ d >= 1, d divides m} (-1)^(m-d)*d^(n-m) = sigma^*_(n-m)(m), n >= 1, m = 1,2, ..., n. For the definition of
sigma^*_(k)(n) see the Hardy reference or a comment in A279395.
O.g.f triangle T: G(z, x) = Sum_{m>=0}
G(m, z)*(x*z)^m, with the column o.g.f. G( m, z) (with offset 0) given in a comment above.

A321543 a(n) = Sum_{d|n} (-1)^(d-1)*d^2.

Original entry on oeis.org

1, -3, 10, -19, 26, -30, 50, -83, 91, -78, 122, -190, 170, -150, 260, -339, 290, -273, 362, -494, 500, -366, 530, -830, 651, -510, 820, -950, 842, -780, 962, -1363, 1220, -870, 1300, -1729, 1370, -1086, 1700, -2158, 1682, -1500, 1850, -2318, 2366, -1590, 2210, -3390, 2451, -1953, 2900, -3230, 2810, -2460, 3172

Offset: 1

N. J. A. Sloane, Nov 23 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Apart from signs, same as A064027.
Glaisher's zeta_i, i=0..12: A048272, A002129, A321543, A138503, A279395, A321544, A321545, A321546, A321547, A321548, A321549, A321550, A321551.
Cf. A321552 - A321565, A321807 - A321836 for similar sequences.

with(numtheory):
a := n -> add( (-1)^(d-1)*d^2, d in divisors(n) ): seq(a(n), n = 1..40);
#  Peter Bala, Jan 11 2021

f[p_, e_] := (p^(2*e + 2) - 1)/(p^2 - 1); f[2, e_] := 2 - (2^(2*e + 2) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 55] (* Amiram Eldar, Nov 04 2022 *)

apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^2), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} (-1)^(k-1)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018
G.f.: Sum_{n >= 1} x^n*(1 - x^n)/(1 + x^n)^3. - Peter Bala, Jan 11 2021
Multiplicative with a(2^e) = 2 - (2^(2*e + 2) - 1)/3, and a(p^e) = (p^(2*e + 2) - 1)/(p^2 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A138503 a(n) = Sum_{d|n} (-1)^(d-1)*d^3.

Original entry on oeis.org

1, -7, 28, -71, 126, -196, 344, -583, 757, -882, 1332, -1988, 2198, -2408, 3528, -4679, 4914, -5299, 6860, -8946, 9632, -9324, 12168, -16324, 15751, -15386, 20440, -24424, 24390, -24696, 29792, -37447, 37296, -34398, 43344, -53747, 50654, -48020, 61544, -73458

Offset: 1

Michael Somos, Mar 21 2008

Also, expansion of (1 - phi(-q)^8) / 16 in powers of q where phi() is a Ramanujan theta function.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q - 7*q^2 + 28*q^3 - 71*q^4 + 126*q^5 - 196*q^6 + 344*q^7 - 583*q^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Index entries for sequences mentioned by Glaisher.

Divisor sums Sum_{d|n} (-1)^(d-1)*d^k: A048272 (k = 0), A002129 (k = 1), A321543 (k = 2), A279395 (k = 4, unsigned), A321544 - A321551 (k = 5 to k = 12).

Cf. A008457, A035016.

with(numtheory):
a := n -> add( (-1)^(d-1)*d^3, d in divisors(n) ): seq(a(n), n = 1..40);
#  Peter Bala, Jan 11 2021

a[ n_] := If[ n < 0, 0, DivisorSum[ n, -(-1)^# #^3&]]; (* Michael Somos, Sep 25 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, q]^8) / 16, {q, 0, n}]; (* Michael Somos, Sep 25 2015 *)
nmax = 40; Rest[CoefficientList[Series[-Product[((1-q^k)/(1+q^k))^8, {k, 1, nmax}]/16, {q, 0, nmax}], q]] (* Vaclav Kotesovec, Sep 26 2015 *)
f[p_, e_] := (p^(3*e + 3) - 1)/(p^3 - 1); f[2, e_] := 2 - (2^(3*e + 3) - 1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 04 2022 *)

{a(n) = if( n<0, 0, sumdiv(n, d, -(-1)^d * d^3))};

Expansion of (1 - (eta(q)^2 / eta(q^2))^8) / 16 in powers of q.
a(n) is multiplicative with a(2^e) = -(8^(e+1) - 15) / 7, a(p^e) = ((p^3)^(e+1) - 1) / (p^3 - 1).
G.f.: Sum_{k>0} k^3 * -(-x)^k / (1 - x^k).
a(n) = -(-1)^n * A008457(n). -16 * a(n) = A035016(n) unless n=0.
G.f.: Sum_{n >= 1} x^n*(1 - 4*x^n + x^(2*n))/(1 + x^n)^4. - Peter Bala, Jan 11 2021

Simpler definition from N. J. A. Sloane, Nov 23 2018

A321544 a(n) = Sum_{d|n} (-1)^(d-1)*d^5.

Original entry on oeis.org

1, -31, 244, -1055, 3126, -7564, 16808, -33823, 59293, -96906, 161052, -257420, 371294, -521048, 762744, -1082399, 1419858, -1838083, 2476100, -3297930, 4101152, -4992612, 6436344, -8252812, 9768751, -11510114, 14408200, -17732440, 20511150, -23645064, 28629152, -34636831, 39296688, -44015598

Offset: 1

N. J. A. Sloane, Nov 23 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Divisor sums Sum_{d|n} (-1)^(d-1)*d^k: A048272 (k = 0), A002129 (k = 1), A321543 (k = 2), A138503 (k = 3), A279395 (k = 4, unsigned), A321545 - A321551 (k = 6 to k = 12).

Cf. A321552 - A321565, A321807 - A321836 for similar sequences.

with(numtheory):
a := n -> add( (-1)^(d-1)*d^5, d in divisors(n) ): seq(a(n), n = 1..40);
#  Peter Bala, Jan 11 2021

f[p_, e_] := (p^(5*e + 5) - 1)/(p^5 - 1); f[2, e_] := 2 - (2^(5*e + 5) - 1)/31; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 35] (* Amiram Eldar, Nov 04 2022 *)

apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^5), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} (-1)^(k-1)*k^5*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018
G.f.: Sum_{n >= 1} x^n*(x^(4*n) - 26*x^(3*n) + 66*x^(2*n) - 26*x^n + 1)/(1 + x^n)^6 (note [1,26,66,26,1] is row 5 of A008292). - Peter Bala, Jan 11 2021
Multiplicative with a(2^e) = 2 - (2^(5*e + 5) - 1)/31, and a(p^e) = (p^(5*e + 5) - 1)/(p^5 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A338549 a(n) = n^4 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^4.

Original entry on oeis.org

1, -17, 80, -240, 624, -1360, 2400, -3840, 6480, -10608, 14640, -19200, 28560, -40800, 49920, -61440, 83520, -110160, 130320, -149760, 192000, -248880, 279840, -307200, 390000, -485520, 524880, -576000, 707280, -848640, 923520, -983040, 1171200, -1419840, 1497600, -1555200

Offset: 1

Ilya Gutkovskiy, Nov 02 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008683, A059377, A279395, A325596, A338547, A338548.

Table[n^4 Sum[(-1)^(n/d + 1) MoebiusMu[d]/d^4, {d, Divisors[n]}], {n, 1, 36}]
nmax = 36; CoefficientList[Series[Sum[MoebiusMu[k] x^k (1 - 11 x^k + 11 x^(2 k) - x^(3 k))/(1 + x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := (p^4 - 1)*p^(4*(e - 1)); f[2, 1] = -17; f[2, e_] := -15*2^(4*(e - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

a(n) = n^4 * sumdiv(n, d, (-1)^(n/d+1)*moebius(d)/d^4); \\ Michel Marcus, Nov 02 2020

G.f.: Sum_{k>=1} mu(k) * x^k * (1 - 11*x^k + 11*x^(2*k) - x^(3*k)) / (1 + x^k)^5.
G.f. A(x) satisfies: A(x) = x * (1 - 11*x + 11*x^2 - x^3) / (1 + x)^5 - Sum_{k>=2} A(x^k).
Dirichlet g.f.: (1 - 2^(5 - s)) * zeta(s - 4) / zeta(s).
a(n) = J_4(n) if n odd, J_4(n) - 32 * J_4(n/2) if n even, where J_4 = A059377 (Jordan function J_4).
Multiplicative with a(2) = -17, a(2^e) = -15*2^(4*(e-1)) for e > 1, and a(p^e) = (p^4-1)*p^(4*(e-1)) for p > 2. - Amiram Eldar, Nov 15 2022

A279397 Row sums of triangle A279396.

Original entry on oeis.org

1, 1, 4, 9, 25, 69, 217, 745, 2801, 11399, 49859, 232639, 1151397, 6017345, 33084489, 190780154, 1150686536, 7241926820, 47455774001, 323158817943, 2282794174761, 16700943712871, 126356690691651, 987303203599953, 7957130740261570, 66071751478724210, 564631175841553564, 4961038559732049506, 44775183035039043106

Offset: 1

Wolfdieter Lang, Jan 10 2017

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Cf. A279395, A279396.

a(n) = Sum_{m=1..n} A279396(n, m) =

Sum_{m=1..n} sigma^*_{n-m}(m), n >= 1, with the definition of sigma^*_k(n) given in the Hardy reference or in a comment in A279395.

cp's OEIS Frontend

A279396 Triangle read by rows T(n, m) = sigma^*(n-m)(m), n >= 1, m = 1, 2, ..., n, with sigma^*(k)(n) given in a comment in A279395.

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A321543 a(n) = Sum_{d|n} (-1)^(d-1)*d^2.

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A138503 a(n) = Sum_{d|n} (-1)^(d-1)*d^3.

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A321544 a(n) = Sum_{d|n} (-1)^(d-1)*d^5.

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A338549 a(n) = n^4 * Sum_{d|n} (-1)^(n/d + 1) * mu(d) / d^4.

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A279397 Row sums of triangle A279396.

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A279396 Triangle read by rows T(n, m) = sigma^(n-m)(m), n >= 1, m = 1, 2, ..., n, with sigma^(k)(n) given in a comment in A279395.