A191832 -id:A191832 - 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

A191822 Number of solutions to the Diophantine equation x1x2 + x2x3 + x3x4 + x4x5 = n, with all xi >= 1.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 8, 16, 20, 32, 36, 58, 58, 86, 92, 125, 122, 178, 164, 228, 224, 286, 268, 382, 330, 436, 424, 534, 474, 660, 556, 740, 692, 840, 752, 1043, 846, 1094, 1032, 1276, 1078, 1476, 1204, 1582, 1458, 1710, 1480, 2070, 1628, 2096, 1924, 2332, 1946, 2652, 2148, 2770, 2480, 2908, 2480, 3512

Offset: 1

N. J. A. Sloane, Jun 17 2011

nonn

Related to "Liouville's Last Theorem".

			G.f.: x^4 + 2 x^5 + 6 x^6 + 8 x^7 + 16 x^8 + 20 x^9 + 32 x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..1000
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_4(n).
E. T. Bell, The form wx+xy+yz+zu, Bull. Amer. Math. Soc., 42 (1936), 377-380.

Cf. A000005, A002133, A065608, A191832.
Cf. A001157, A038040, A055507.
Cf. A189835.

with(numtheory);
D00:=n->add(tau(j)*tau(n-j),j=1..n-1);
L4:=n->sigma[2](n)-n*sigma[0](n)-D00(n);
[seq(L4(n),n=1..60)];

a[ n_] := Length @ FindInstance[{x1 > 0, x2 > 0, x3 > 0, x4 > 0, x5 > 0, n == x1 x2 + x2 x3 + x3 x4 + x4 x5}, {x1, x2, x3, x4, x5}, Integers, 10^9]; (* Michael Somos, Nov 12 2016 *)

a(n) = sigma_2(n) - n*sigma_0(n) - A055507(n-1).

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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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A191822 Number of solutions to the Diophantine equation x1x2 + x2x3 + x3x4 + x4x5 = 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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A191822 Number of solutions to the Diophantine equation x1*x2 + x2*x3 + x3*x4 + x4*x5 = n, with all xi >= 1.

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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) ).

A191822 Number of solutions to the Diophantine equation x1x2 + x2x3 + x3x4 + x4x5 = n, with all xi >= 1.