cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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) ).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Feb 24 2001

Keywords

Links

Crossrefs

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)).

Programs

  • Maple
    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
  • PARI
    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

Formula

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

A374951 a(n) = Sum_{i+j+k=n, i,j,k >= 1} sigma(i) * sigma(j) * sigma(k).

Original entry on oeis.org

0, 0, 1, 9, 39, 120, 300, 645, 1261, 2262, 3825, 6160, 9471, 14178, 20376, 28965, 39600, 54066, 71145, 94248, 120140, 155310, 193116, 244560, 297819, 370860, 443710, 544554, 641655, 778458, 904800, 1085445, 1248762, 1483308, 1688052, 1991515, 2244375, 2626380
Offset: 1

Views

Author

Seiichi Manyama, Jul 25 2024

Keywords

Links

Crossrefs

Column k=3 of A319083.
Cf. A000203, A000385, A191829.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
       add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 3):
seq(a(n), n=1..55);  # Alois P. Heinz, Jul 25 2024
  • Mathematica
    b[n_, k_] := b[n, k] = If[k == 0, If[n == 0, 1, 0],
   If[k == 1, If[n == 0, 0, DivisorSigma[1, n]], Function[q,
   Sum[b[j, q]*b[n - j, k - q], {j, 0, n}]][Quotient[k, 2]]]];
a[n_] := b[n, 3];
Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Mar 14 2025, after Alois P. Heinz *)
  • PARI
    my(N=40, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, k*x^k/(1-x^k))^3))
  • Python
    from sympy import divisor_sigma
def A374951(n): return (60*sum(divisor_sigma(i)*divisor_sigma(n-i,3) for i in range(1,n))+divisor_sigma(n)*(9*n*(2*n-1)+1)-5*divisor_sigma(n,3)*(3*n-1))//144  # Chai Wah Wu, Jul 25 2024

Formula

G.f.: ( Sum_{k>=1} k * x^k/(1 - x^k) )^3 = ( Sum_{k>=1} x^k/(1 - x^k)^2 )^3.
a(n) = Sum_{i=1..n-2} sigma(i)*A000385(n-i-1). - Chai Wah Wu, Jul 25 2024
Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / 155520. - Vaclav Kotesovec, Sep 19 2024

A191832 Number of solutions to the Diophantine equation x1*x2 + x2*x3 + x3*x4 + x4*x5 + 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

Views

Author

N. J. A. Sloane, Jun 17 2011

Keywords

Comments

Related to "Liouville's Last Theorem".

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

A320019 Coefficients of polynomials related to the number of divisors, triangle read by rows, T(n,k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 4, 1, 0, 3, 8, 6, 1, 0, 2, 14, 18, 8, 1, 0, 4, 20, 41, 32, 10, 1, 0, 2, 28, 78, 92, 50, 12, 1, 0, 4, 37, 132, 216, 175, 72, 14, 1, 0, 3, 44, 209, 440, 490, 298, 98, 16, 1, 0, 4, 58, 306, 814, 1172, 972, 469, 128, 18, 1
Offset: 0

Views

Author

Peter Luschny, Oct 03 2018

Keywords

Comments

Column k is the k-fold self-convolution of tau (A000005). - Alois P. Heinz, Feb 01 2021

Examples

			Triangle starts:
[0] 1
[1] 0, 1
[2] 0, 2,  1
[3] 0, 2,  4,   1
[4] 0, 3,  8,   6,   1
[5] 0, 2, 14,  18,   8,   1
[6] 0, 4, 20,  41,  32,  10,   1
[7] 0, 2, 28,  78,  92,  50,  12,  1
[8] 0, 4, 37, 132, 216, 175,  72, 14,  1
[9] 0, 3, 44, 209, 440, 490, 298, 98, 16, 1

Links

Crossrefs

Columns k=0-4 give: A000007, A000005, A055507, A191829, A375002.
Row sums are A129921.
T(2n,n) gives A340992.
Cf. A319083.

Programs

  • Maple
    P := proc(n, x) option remember; if n = 0 then 1 else
x*add(numtheory:-tau(n-k)*P(k,x), k=0..n-1) fi end:
Trow := n -> seq(coeff(P(n, x), x, k), k=0..n):
seq(lprint([n], Trow(n)), n=0..9);
# second Maple program:
T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[tau](n)), (q->
       add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 01 2021
# Uses function PMatrix from A357368.
PMatrix(10, NumberTheory:-tau); # Peter Luschny, Oct 19 2022
  • Mathematica
    T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
     If[k == 1, If[n == 0, 0, DivisorSigma[0, n]],
     With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n-j, k-q], {j, 0, n}]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 11 2021, after Alois P. Heinz *)

Formula

The polynomials are defined by recurrence: p(0,x) = 1 and for n > 0 by
p(n, x) = x*Sum_{k=0..n-1} tau(n-k)*p(k, x).
Sigma[k](n) computes the sum of the k-th power of positive divisors of n. The recurrence applied with k = 0 gives this triangle, with k = 1 gives A319083.
T(n,k) = [x^n] (Sum_{j>=1} tau(j)*x^j)^k. - Alois P. Heinz, Feb 14 2021

A341374 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: [Sum_{n>=0} x^n/(1 - x^(n+1))]^3 = Sum_{n>=0} a(n)*x^n/(1 - x^(n+1))^3.

Original entry on oeis.org

1, 3, 12, 22, 63, 57, 181, 174, 318, 302, 714, 444, 1177, 852, 1239, 1349, 2598, 1440, 3586, 2226, 3381, 3282, 6246, 3174, 6980, 5343, 7434, 6031, 12111, 5076, 14638, 9636, 12381, 11513, 16125, 9441, 24115, 15765, 19743, 14982, 32076, 13317, 36726, 21783, 25062
Offset: 0

Views

Author

Paul D. Hanna, Feb 11 2021

Keywords

Examples

			A(x) = 1 + 3*x + 12*x^2 + 22*x^3 + 63*x^4 + 57*x^5 + 181*x^6 + 174*x^7 + 318*x^8 + 302*x^9 + 714*x^10 + 444*x^11 + 1177*x^12 + ...
such that
D(x)^3 = 1/(1-x)^3 + 3*x/(1-x^2)^3 + 12*x^2/(1-x^3)^3 + 22*x^3/(1-x^4)^3 + 63*x^4/(1-x^5)^3 + 57*x^5/(1-x^6)^3 + ... + a(n)*x^n/(1-x^(n+1))^3 + ...
and
D(x)^3 = A(x) + 3*x*A(x^2) + 6*x^2*A(x^3) + 10*x^3*A(x^4) + 15*x^4*A(x^5) + 21*x^5*A(x^6) + 28*x^6*A(x^7) + ... + (n+1)*(n+2)/2*x^n*A(x^(n+1)) + ...
where
D(x)^3 = 1 + 6*x + 18*x^2 + 41*x^3 + 78*x^4 + 132*x^5 + 209*x^6 + 306*x^7 + 435*x^8 + 591*x^9 + 780*x^10 + 1008*x^11 + ... + A191829(n+1)*x^n + ...
D(x) = 1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 4*x^7 + 3*x^8 + 4*x^9 + 2*x^10 + 6*x^11 + 2*x^12 + ... + A000005(n+1)*x^n + ...

Crossrefs

Cf. A341373, A341375, A191829, A000005.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(n=0,#A, x^n/(1 - x^(n+1) +x*O(x^#A)) )^3 - sum(n=0,#A-1,A[n+1]*x^n/(1 - x^(n+1) + x*O(x^#A))^3 ), #A-1) );A[n+1]}
for(n=0,100,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [ Sum_{n>=0} x^n/(1 - x^(n+1)) ]^3 = Sum_{n>=0} a(n) * x^n / (1 - x^(n+1))^3.
(2) [ Sum_{n>=0} x^n/(1 - x^(n+1)) ]^3 = Sum_{n>=0} (n+1)*(n+2)/2 * x^n * A( x^(n+1) ).

A350596 Coefficients of the expansion of Sum_{i,j,k>=1} x^(i*j*k)/((1-x^i)*(1-x^j)*(1-x^k)).

Original entry on oeis.org

1, 6, 15, 34, 54, 96, 130, 196, 255, 349, 417, 570, 652, 823, 954, 1180, 1299, 1602, 1732, 2089, 2280, 2659, 2820, 3375, 3541, 4078, 4321, 4963, 5139, 5970, 6115, 6982, 7233, 8116, 8325, 9544, 9634, 10780, 11040, 12385, 12465, 14091, 14071, 15730, 15976, 17596, 17580
Offset: 1

Views

Author

Seiichi Manyama, Jan 08 2022

Keywords

Crossrefs

Cf. A000005, A191829, A274628.

Programs

  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(i=1, N, sum(j=1, N\i, sum(k=1, N\(i*j), x^(i*j*k)/((1-x^i)*(1-x^j)*(1-x^k))))))
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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