A191822 -id:A191822 - cp's OEIS frontend

A059820 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).

0, 1, 4, 9, 19, 30, 52, 70, 107, 136, 191, 226, 314, 352, 463, 523, 664, 717, 919, 964, 1205, 1282, 1546, 1603, 1992, 2009, 2414, 2504, 2958, 2974, 3606, 3553, 4223, 4273, 4936, 4912, 5885, 5685, 6634, 6654, 7664, 7454, 8822, 8454, 9845

Offset: 0

N. J. A. Sloane, Feb 24 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), A059821(k=4), A059822 (k=5), A059823 (k=6), A059824 (k=7), A059825 (k=8).
Cf. A002133, A065608, A191822, A191832.
Cf. A000203, A001157, A055507, A191829 (Andrews's D_{0,0,0}(n)), A191831 (Andrews's D_{0,1}(n)).

Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=3

D(x, y, n) = sum(k=1, n-1, sigma(k, x)*sigma(n-k, y));
D000(n) = sum(k=1, n-1, sigma(k, 0)*D(0, 0, n-k));
a(n) = if(n==0, 0, (3*D(0, 0, n)+3*D(0, 1, n)+D000(n)+2*sigma(n, 0)+3*sigma(n)+sigma(n, 2))/6); \\ Seiichi Manyama, Jul 26 2024

a(n) = ( 3*A055507(n-1) + 3*A191831(n) + A191829(n) + 2*sigma_0(n) + 3*sigma(n) + sigma_2(n) )/6. - Seiichi Manyama, Jul 26 2024

A189835 Number of representations of n as ab + bc + cd + de where a, b, d, e>0, c>=0 are integers.

Original entry on oeis.org

0, 1, 4, 9, 16, 26, 36, 53, 64, 90, 100, 138, 144, 194, 200, 261, 256, 347, 324, 426, 416, 522, 484, 658, 576, 746, 712, 882, 784, 1060, 900, 1173, 1088, 1314, 1160, 1587, 1296, 1658, 1544, 1890, 1600, 2164, 1764, 2298, 2096, 2466, 2116, 2930, 2304, 2955, 2696

Offset: 1

Michael Somos, Apr 28 2011

nonn

Related to "Liouville's Last Theorem".

Inverse Möbius transform of A344485(n). - Wesley Ivan Hurt, Jul 16 2025

			G.f. = x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 26*x^6 + 36*x^7 + 53*x^8 + 64*x^9 + 90*x^10 + ...
a(3) = 4 since 3 = 1*1 + 1*0 + 0*1 + 1*2 = 1*1 + 1*0 + 0*2 + 2*1 = 1*2 + 2*0 + 0*1 + 1*1 = 2*1 + 1*0 + 0*1 + 1*1 are all 4 representations of 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293. see p. 273.
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_5(n).
E. T. Bell, The form wx+xy+yz+zu, Bull. Amer. Math. Soc., 42 (1936), 377-380.

Cf. A001157, A038040, A191822, A344485.

a189835 n = a001157 n - a038040 n  -- Reinhard Zumkeller, Jan 21 2014

with(numtheory); f:=n->sigma[2](n)-n*sigma[0](n);

a[n_] := DivisorSigma[2, n] - n*DivisorSigma[0, n]; Table[a[n], {n, 51}] (* Jean-François Alcover, Aug 31 2011 *)

{a(n) = if( n<1, 0, sigma( n, 2) - n * sigma( n, 0))}

G.f.: Sum_{k>0} (x^k + x^(2*k)) / (1 - x^k)^3 - k * x^k / (1 - x^k)^2.
a(n) = A001157(n) - A038040(n) = sigma(n,2) - n*sigma(n,0) where sigma(n,k) is the sum of the k-th powers of the divisors of n.
a(n) = Sum_{d|n} A344485(d). - Wesley Ivan Hurt, Jul 16 2025

Added references, comment, Maple program, cross-reference to A191822. - N. J. A. Sloane, Jun 17 2011

A191832 Number of solutions to the Diophantine equation x1x2 + x2x3 + x3x4 + x4x5 + x5*x6 = n, with all xi >= 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 7, 10, 22, 29, 51, 61, 99, 115, 163, 192, 262, 287, 385, 428, 528, 600, 730, 780, 963, 1054, 1202, 1337, 1545, 1646, 1908, 2059, 2269, 2516, 2770, 2933, 3298, 3568, 3792, 4142, 4493, 4786, 5183, 5562, 5831, 6423, 6745, 7140, 7639, 8231, 8479, 9216, 9603, 10260, 10663, 11488, 11752, 12838, 13100, 13887

Offset: 1

N. J. A. Sloane, Jun 17 2011

nonn

Related to "Liouville's Last Theorem".

Robert Israel, Table of n, a(n) for n = 1..1000
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_5(n).

Cf. A000005, A000203, A002133, A055507, A191822, A191829, A191831.

with(numtheory);
D00:=n->add(tau(j)*tau(n-j),j=1..n-1);
D01:=n->add(tau(j)*sigma(n-j),j=1..n-1);
D000:=proc(n) local t1,i,j;
t1:=0;
for i from 1 to n-1 do
for j from 1 to n-1 do
if (i+j < n) then t1 := t1+numtheory:-tau(i)*numtheory:-tau(j)*numtheory:-tau(n-i-j); fi;
od; od;
t1;
end;
L5:=n->D000(n)/6+D00(n)+D01(n)/2+(2*n-1/6)*tau(n)-11*sigma[2](n)/6;
[seq(L5(n),n=1..60)];
# Alternate:
g:= proc(n,k,j) option remember;
     if n < k-1 then 0
     elif k = 2 then
        if n mod j = 0 then 1 else 0 fi
     else
        add(procname(n-j*x,k-1,x), x=1 .. floor((n-k+2)/j))
     fi
end proc:
f:= n -> add(g(n,6,j),j=1..n-4);
seq(f(n),n=1..100); # Robert Israel, Dec 02 2015

g[n_, k_, j_] := g[n, k, j] = If[n < k - 1, 0, If[k == 2, If[ Mod[n, j] == 0, 1, 0], Sum[g[n - j x, k - 1, x], {x, 1, Floor[(n - k + 2)/j]}]]];
f[n_] := Sum[g[n, 6, j], {j, 1, n - 4}];
Array[f, 100] (* Jean-François Alcover, Sep 25 2020, after Robert Israel *)

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A059820 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).

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A189835 Number of representations of n as ab + bc + cd + de where a, b, d, e>0, c>=0 are integers.

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A191832 Number of solutions to the Diophantine equation x1x2 + x2x3 + x3x4 + x4x5 + x5*x6 = n, with all xi >= 1.

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cp's OEIS Frontend

A059820 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).

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A189835 Number of representations of n as a*b + b*c + c*d + d*e where a, b, d, e>0, c>=0 are integers.

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A191832 Number of solutions to the Diophantine equation x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x6 = n, with all xi >= 1.

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Author

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Maple

Mathematica

A059820 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).

A189835 Number of representations of n as ab + bc + cd + de where a, b, d, e>0, c>=0 are integers.

A191832 Number of solutions to the Diophantine equation x1x2 + x2x3 + x3x4 + x4x5 + x5*x6 = n, with all xi >= 1.