A162552 -id:A162552 - cp's OEIS frontend

A162553 G.f.: A(x) = exp( Sum_{n>=1} A162552(n)^2*x^n/n ) where the l.g.f. of A162552 is the log of the characteristic function of the squares.

1, 1, 1, 1, 3, 6, 10, 15, 18, 35, 73, 143, 230, 296, 416, 753, 1673, 2934, 4203, 5654, 9135, 17881, 33102, 52787, 73749, 107869, 189629, 359107, 619296, 923833, 1306855, 2065717, 3776424, 6823452, 10935160, 15822727, 23395694, 39675378

Offset: 0

Paul D. Hanna, Jul 06 2009

nonn

A162552 is defined by: exp( Sum_{n>=1} A162552(n)*x^n/n ) = Sum_{n>=0} x^(n^2).

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 15*x^6 +...
log(A(x)) = x + x^2/2 + x^3/3 + 9*x^4/4 + 16*x^5/5 + 25*x^6/6 + 36*x^7/7 +...+ A162552(n)^2*x^n/n +...
Let L(x) = x - 1*x^2/2 + 1*x^3/3 + 3*x^4/4 - 4*x^5/5 + 5*x^6/6 - 6*x^7/7 +...+ A162552(n)*x^n/n +... then
exp(L(x)) = 1 + x + x^4 + x^9 + x^16 + x^25 + x^36 +...+ x^(n^2) +...
is the characteristic function of the squares (A010052).

Paul D. Hanna, Table of n, a(n), n = 0..330.

Cf. A162552, A010052, A162416 (variant).

{a(n)=local(Q=sum(m=0,n,x^(m^2))+x*O(x^n),A); A=exp(sum(k=1,n,polcoeff(log(Q),k)^2*k*x^k)+x*O(x^n));polcoeff(A,n)}

A161808 G.f.: A(q) = exp( Sum_{n>=1} A162552(n) * 3A038500(n) q^n/n ).

Original entry on oeis.org

1, 3, 3, 3, 9, 12, 12, 27, 36, 57, 141, 165, 135, 321, 450, 399, 780, 1068, 1308, 2913, 3537, 2736, 5940, 8430, 7173, 13251, 18267, 17661, 35007, 45051, 31866, 58506, 85890, 65694, 102000, 145293, 101547, 140574, 203781, 114765, 93051, 161754

Offset: 0

Paul D. Hanna, Jul 21 2009

sign

A162552 forms the l.g.f. of log[ Sum_{n>=0} x^(n^2) ], and
A038500(n) is the highest power of 3 dividing n.
The first negative term is a(43) = -162729.

			G.f.: A(q) = 1 + 3*q + 3*q^2 + 3*q^3 + 9*q^4 + 12*q^5 + 12*q^6 +...
log(A(q)) = 3*q - 3*q^2/2 + 9*q^3/3 + 9*q^4/4 - 12*q^5/5 + 45*q^6/6 - 18*q^7/7 +...
Compare to: q - q^2/2 + q^3/3 + 3*q^4/4 - 4*q^5/5 + 5*q^6/6 - 6*q^7/7 +...
which equals log( Sum_{n>=0} q^(n^2) ) as described by A162552.

Paul D. Hanna, Table of n, a(n) for n = 0..10000 (terms 0..100 from Georg Fischer)

Cf. A161804 (variant).

{a(n)=local(Q=sum(m=0,n,x^(m^2))+x*O(x^n),A); A=exp(sum(k=1,n,polcoeff(log(Q),k)*3*3^valuation(k,3)*x^k)+x*O(x^n));polcoeff(A,n)}

A161803 G.f.: A(x) = exp( Sum_{n>=1} A162552(n) * 2A006519(n) x^n/n ).

Original entry on oeis.org

1, 2, 0, -2, 6, 12, 0, -8, 24, 44, 0, -30, 54, 104, 0, -60, 238, 466, 0, -402, 924, 1892, 0, -1228, 3264, 6006, 0, -4052, 6688, 13052, 0, -7452, 16536, 32140, 0, -24828, 39660, 85744, 0, -53592, 114336, 212406, 0, -141090, 190754, 386956, 0, -216572, 136078

Offset: 0

Paul D. Hanna, Jul 19 2009

sign

A162552 forms the l.g.f. of log[ Sum_{n>=0} x^(n^2) ], while
2*A006519 forms the l.g.f. of binary partitions (A000123) and
A006519(n) is the highest power of 2 dividing n.

			G.f.: 1 + 2*x - 2*x^3 + 6*x^4 + 12*x^5 - 8*x^7 + 24*x^8 + 44*x^9 +...

Cf. A161800, A162552.

{a(n)=local(SQ=sum(m=0, sqrtint(n+1), x^(m^2))+x*O(x^n), L=sum(m=1,n,2*2^valuation(m,2)*polcoeff(log(SQ),m)*x^m)+x*O(x^n)); polcoeff(exp(L),n)}

A363783 L.g.f.: log( Sum_{k>=0} x^(k^3) ).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, 7, -8, 9, -10, 11, -12, 13, -14, 7, 1, -10, 20, -31, 43, -56, 70, -77, 76, -66, 73, -43, 1, 54, -123, 199, -274, 339, -419, 470, -480, 436, -324, 137, 124, -449, 861, -1332, 1822, -2278, 2633, -2813, 2745, -2366, 1582, -326, -1430, 3635, -6225, 9037, -11836, 14325, -16106, 16706, -15615

Offset: 1

Seiichi Manyama, Jun 21 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A162552, A363784.

L.g.f.: L(x) = Sum_{k>=1} a(k)*x^k/k = log( Sum_{k>=0} x^(k^3) ).

A363778 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(Sum_{j>=0} x^(j^2))^k.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 3, -1, 0, 1, -4, 6, -4, 0, 0, 1, -5, 10, -10, 3, 1, 0, 1, -6, 15, -20, 12, 0, -2, 0, 1, -7, 21, -35, 31, -9, -5, 3, 0, 1, -8, 28, -56, 65, -36, -2, 12, -3, 0, 1, -9, 36, -84, 120, -96, 24, 24, -18, 1, 0, 1, -10, 45, -120, 203, -210, 105, 20, -54, 18, 2, 0

Offset: 0

Seiichi Manyama, Jun 21 2023

			Square array begins:
  1,  1,  1,   1,   1,   1,    1, ...
  0, -1, -2,  -3,  -4,  -5,   -6, ...
  0,  1,  3,   6,  10,  15,   21, ...
  0, -1, -4, -10, -20, -35,  -56, ...
  0,  0,  3,  12,  31,  65,  120, ...
  0,  1,  0,  -9, -36, -96, -210, ...
  0, -2, -5,  -2,  24, 105,  294, ...

Columns k=0..3 give A000007, A317665, A363774, A363775.
Main diagonal gives A363780.
Cf. A045847, A162552, A363779.

T(0,k) = 1; T(n,k) = -(k/n) * Sum_{j=1..n} A162552(j) * T(n-j,k).

A216273 Triangle generated by Sum_{n>=1, k=1..n} T(n,k)x^ny^k/n = log(1 + Sum_{n>=1} y*x^(n^2)), where coefficients are read by rows.

Original entry on oeis.org

1, 0, -1, 0, 0, 1, 4, 0, 0, -1, 0, -5, 0, 0, 1, 0, 0, 6, 0, 0, -1, 0, 0, 0, -7, 0, 0, 1, 0, -4, 0, 0, 8, 0, 0, -1, 9, 0, 9, 0, 0, -9, 0, 0, 1, 0, -10, 0, -15, 0, 0, 10, 0, 0, -1, 0, 0, 11, 0, 22, 0, 0, -11, 0, 0, 1, 0, 0, 4, -12, 0, -30, 0, 0, 12, 0, 0, -1, 0, -13, 0, -13, 13, 0, 39, 0, 0, -13, 0, 0, 1, 0, 0, 28, 0, 28, -14, 0, -49, 0, 0, 14, 0, 0, -1

Offset: 1

Paul D. Hanna, Mar 16 2013

			G.f.: A(x,y) = y*x - y^2*x^2/2 + y^3*x^3/3 + (-y^4 + 4*y)*x^4/4 + (y^5 - 5*y^2)*x^5/5 + (-y^6 + 6*y^3)*x^6/6 + (y^7 - 7*y^4)*x^7/7 + (-y^8 + 8
*y^5 - 4*y^2)*x^8/8 + (y^9 - 9*y^6 + 9*y^3 + 9*y)*x^9/9 + (-y^10 + 10*y^7 - 15*y^4 - 10*y^2)*x^10/10 +...
where
exp(A(x,y)) = 1 + y*x + y*x^4 + y*x^9 + y*x^16 + y*x^25 +...
Triangle begins:
1;
0, -1;
0, 0, 1;
4, 0, 0, -1;
0, -5, 0, 0, 1;
0, 0, 6, 0, 0, -1;
0, 0, 0, -7, 0, 0, 1;
0, -4, 0, 0, 8, 0, 0, -1;
9, 0, 9, 0, 0, -9, 0, 0, 1;
0, -10, 0, -15, 0, 0, 10, 0, 0, -1;
0, 0, 11, 0, 22, 0, 0, -11, 0, 0, 1;
0, 0, 4, -12, 0, -30, 0, 0, 12, 0, 0, -1;
0, -13, 0, -13, 13, 0, 39, 0, 0, -13, 0, 0, 1;
0, 0, 28, 0, 28, -14, 0, -49, 0, 0, 14, 0, 0, -1;
0, 0, 0, -45, 0, -50, 15, 0, 60, 0, 0, -15, 0, 0, 1;
16, 0, 0, -4, 64, 0, 80, -16, 0, -72, 0, 0, 16, 0, 0, -1;
0, -17, 17, 0, 17, -85, 0, -119, 17, 0, 85, 0, 0, -17, 0, 0, 1;
0, -9, 18, -54, 0, -45, 108, 0, 168, -18, 0, -99, 0, 0, 18, 0, 0, -1;
0, 0, 19, -19, 114, 0, 95, -133, 0, -228, 19, 0, 114, 0, 0, -19, 0, 0, 1;
0, -20, 0, -30, 24, -200, 0, -175, 160, 0, 300, -20, 0, -130, 0, 0, 20, 0, 0, -1;
0, 0, 42, -21, 42, -42, 315, 0, 294, -189, 0, -385, 21, 0, 147, 0, 0, -21, 0, 0, 1;
0, 0, 22, -66, 88, -55, 88, -462, 0, -462, 220, 0, 484, -22, 0, -165, 0, 0, 22, 0, 0, -1;
0, 0, 0, -69, 92, -230, 69, -184, 644, 0, 690, -253, 0, -598, 23, 0, 184, 0, 0, -23, 0, 0, 1;
0, 0, 24, 0, 144, -124, 480, -84, 360, -864, 0, -990, 288, 0, 728, -24, 0, -204, 0, 0, 24, 0, 0, -1;
25, -25, 0, -75, 25, -250, 175, -875, 100, -655, 1125, 0, 1375, -325, 0, -875, 25, 0, 225, 0, 0, -25, 0, 0, 1; ...

Paul D. Hanna, Rows n=1..45, flattened as a list of n, a(n) for n = 1..1035.

Cf. A227311, A162552, A000203, A015128, A002318, A004404, A004405.

{T(n,k)=n*polcoeff(polcoeff(log(1+sum(m=1,sqrtint(n)+1,y*x^(m^2))+x*O(x^n)),n,x),k,y)}
for(n=1,25,for(k=1,n,print1(T(n,k),", "));print(""))

/* Alternate g.f., true for all m >= 0: */
{T(n,k,m=0) = if(k<1||m<0,0, (n/k/binomial(k+m,m)) * polcoeff(polcoeff( 1 - 1/(1+sum(j=1,sqrtint(n+1),y*x^(j^2))+x*O(x^n))^(m+1), n,x),k,y))}
for(n=1, 25, for(k=1, n, print1(T(n, k, 1), ", ")); print(""))

G.f.: Sum_{n>=1, k=1..n} T(n,k)*x^n*y^k*k*binomial(k+m,m)/n = 1 - 1/(1 + Sum_{n>=1} y*x^(n^2))^(m+1), which holds for all m >= 0.
Row sums equal A162552.
Sum_{k=1..n} T(n,k)*2^k = -(-1)^n*(sigma(2*n) - sigma(n)) for n>=1, where sigma is the sum of divisors of n, A000203.
Sum_{k=1..n} T(n,k)*2^k*k = -(-1)^n*n*A015128(n) for n>=1, where A015128(n) is the number of overpartitions of n, with g.f.: Product_{n>=1} (1+x^n)/(1-x^n).
Sum_{k=1..n} T(n,k)*2^k*k*(k+1) = -(-1)^n*4*n*A002318(n) for n>=1, where A002318 lists the coefficients in (1/theta_4(q)^2 -1)/4 in powers of q.
Sum_{k=1..n} T(n,k)*2^k*k*(k+1)*(k+2)/2! = -n*A004404(n) for n>=1, where A004404 lists the coefficients in 1/(1 + Sum_{n>=1} 2*x^(n^2))^3.
Sum_{k=1..n} T(n,k)*2^k*k*(k+1)*(k+2)*(k+3)/3! = -n*A004405(n) for n>=1, where A004405 lists the coefficients in 1/(1 + Sum_{n>=1} 2*x^(n^2))^4.
More generally:
Sum_{k=1..n} T(n,k)*y^k*k*binomial(k+m,m)/n = [x^n] 1 - 1/(1 + Sum_{n>=1} y*x^(n^2))^(m+1) for m>=0, n>=1.

A363784 L.g.f.: log( Sum_{k>=0} x^(k^4) ).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 15, 1, -18, 36, -55, 75, -96, 118, -141, 165, -190, 216, -243, 271, -300, 330, -345, 344, -326, 290, -235, 160, -64, -54, 195, -360, 550, -766, 1009, -1280, 1580

Offset: 1

Seiichi Manyama, Jun 21 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A162552, A363783.

L.g.f.: L(x) = Sum_{k>=1} a(k)*x^k/k = log( Sum_{k>=0} x^(k^4) ).

A305620 Expansion of e.g.f. log(1 + Sum_{k>=1} x^(k^2)/k^2).

Original entry on oeis.org

1, -1, 2, 0, -6, 60, -540, 3780, 12600, -199080, 3074400, -45738000, 511434000, -5621616000, 55394539200, 960323364000, -24001273296000, 498178528848000, -9994137465312000, 156104172544320000, -2076607873660320000, 18061446353670720000, 206725394268993600000

Offset: 1

Ilya Gutkovskiy, Jun 06 2018

sign

			E.g.f.: A(x) = x - x^2/2! + 2*x^3/3! - 6*x^5/5! + 60*x^6/6! - 540*x^7/7! + ...
exp(A(x)) = 1 + x + x^4/4 + x^9/9 + x^16/16 + ... + x^A000290(k)/A000290(k) + ...
exp(exp(A(x))-1) = 1 + x + x^2/2! + x^3/3! + 7*x^4/4! + 31*x^5/5! + ... + A205801(k)*x^k/k! + ... = Product_{j>=1} 1/(1 - x^j)^(A008836(j)/j).

Robert Israel, Table of n, a(n) for n = 1..450

Cf. A000290, A008836, A162552, A205801, A205804.

N:= 50: # for a(1)..a(N)
g:= log(1 + add(x^(k^2)/k^2,k=1..floor(sqrt(N)))):
S:= series(g,x,N+1):
seq(coeff(S,x,n)*n!,n=1..N); # Robert Israel, Jun 07 2018

nmax = 23; Rest[CoefficientList[Series[Log[1 + Sum[x^k^2/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 23; Rest[CoefficientList[Series[Log[1 + Log[Product[1/(1 - x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}]]], {x, 0, nmax}], x] Range[0, nmax]!]
a[n_] := a[n] = Boole[IntegerQ[n^(1/2)]] (n - 1)! - Sum[k Binomial[n, k] Boole[IntegerQ[(n - k)^(1/2)]] (n - k - 1)! a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 23}]

A363774 Expansion of 1/(Sum_{k>=0} x^(k^2))^2.

Original entry on oeis.org

1, -2, 3, -4, 3, 0, -5, 12, -18, 18, -9, -12, 44, -76, 93, -76, 5, 120, -273, 400, -414, 228, 200, -828, 1480, -1842, 1539, -268, -2004, 4824, -7168, 7568, -4518, -2784, 13577, -24900, 31563, -27236, 6816, 30308, -77010, 116844, -126018, 80180, 34140, -205932, 389275

Offset: 0

Seiichi Manyama, Jun 21 2023

Convolution inverse of A000925.
Column k=2 of A363778.
Cf. A162552, A274621.

my(N=50, x='x+O('x^N)); Vec(1/sum(k=0, sqrtint(N), x^k^2)^2)

a(0) = 1; a(n) = -(2/n) * Sum_{k=1..n} A162552(k) * a(n-k).

A363775 Expansion of 1/(Sum_{k>=0} x^(k^2))^3.

Original entry on oeis.org

1, -3, 6, -10, 12, -9, -2, 24, -54, 80, -84, 42, 66, -234, 420, -536, 450, -39, -740, 1770, -2688, 2898, -1722, -1320, 6078, -11349, 14736, -12992, 3084, 15999, -41212, 64032, -70788, 46020, 20778, -126132, 244120, -323421, 295410, -96848, -293868, 815829, -1297972

Offset: 0

Seiichi Manyama, Jun 21 2023

Convolution inverse of A002102.
Column k=3 of A363778.
Cf. A162552.

my(N=50, x='x+O('x^N)); Vec(1/sum(k=0, sqrtint(N), x^k^2)^3)

a(0) = 1; a(n) = -(3/n) * Sum_{k=1..n} A162552(k) * a(n-k).

cp's OEIS Frontend

A162553 G.f.: A(x) = exp( Sum_{n>=1} A162552(n)^2*x^n/n ) where the l.g.f. of A162552 is the log of the characteristic function of the squares.

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A161808 G.f.: A(q) = exp( Sum_{n>=1} A162552(n) * 3*A038500(n) * q^n/n ).

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A161803 G.f.: A(x) = exp( Sum_{n>=1} A162552(n) * 2*A006519(n) * x^n/n ).

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A363783 L.g.f.: log( Sum_{k>=0} x^(k^3) ).

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A363778 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(Sum_{j>=0} x^(j^2))^k.

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A216273 Triangle generated by Sum_{n>=1, k=1..n} T(n,k)*x^n*y^k/n = log(1 + Sum_{n>=1} y*x^(n^2)), where coefficients are read by rows.

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A363784 L.g.f.: log( Sum_{k>=0} x^(k^4) ).

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A305620 Expansion of e.g.f. log(1 + Sum_{k>=1} x^(k^2)/k^2).

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Mathematica

A363774 Expansion of 1/(Sum_{k>=0} x^(k^2))^2.

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A161808 G.f.: A(q) = exp( Sum_{n>=1} A162552(n) * 3A038500(n) q^n/n ).

A161803 G.f.: A(x) = exp( Sum_{n>=1} A162552(n) * 2A006519(n) x^n/n ).

A216273 Triangle generated by Sum_{n>=1, k=1..n} T(n,k)x^ny^k/n = log(1 + Sum_{n>=1} y*x^(n^2)), where coefficients are read by rows.