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.

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A233969 Partial sums of A006950.

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 17, 24, 34, 47, 63, 84, 112, 147, 190, 245, 315, 401, 506, 636, 797, 993, 1229, 1516, 1866, 2286, 2787, 3389, 4111, 4969, 5985, 7191, 8622, 10309, 12290, 14621, 17362, 20568, 24308, 28676, 33772, 39694, 46562, 54529, 63762, 74432, 86738
Offset: 0

Views

Author

Omar E. Pol, Jan 12 2014

Keywords

Comments

The first three columns of A211970 are A211971, A000041, A006950, so for k = 0..2, the partial sums of column k of A211970 give: A015128, A000070, this sequence.

Links

Crossrefs

Cf. A000041, A000070, A015128, A195825, A195826, A195836, A210843, A211970, A211971, A233758, A233759.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i-irem(i, 2)))))
    end:
a:= proc(n) option remember; b(n, n) +`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 12 2014
  • Mathematica
    Accumulate[CoefficientList[Series[x*QPochhammer[-1/x, x^2]/((1 + x) * QPochhammer[x^2]), {x, 0, 50}], x]] (* Vaclav Kotesovec, Oct 27 2016 *)

Formula

a(n) ~ exp(Pi*sqrt(n/2))/(2*Pi*sqrt(n)). - Vaclav Kotesovec, Oct 27 2016

A210992 Square array read by antidiagonals, in which column k starts with k plateaus of lengths k+1, k, k-1, k-2, k-3,..2 and of levels A000124: 1, 2, 4, 7, 11..., if k >= 1, connected by consecutive integers. After the last plateau the length remains 1.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 5, 3, 1, 1, 1, 6, 4, 2, 1, 1, 1, 7, 5, 2, 1, 1, 1, 1, 8, 6, 3, 2, 1, 1, 1, 1, 9, 7, 4, 2, 1, 1, 1, 1, 1, 10, 8, 5, 2, 2, 1, 1, 1, 1, 1, 11, 9, 6, 3, 2, 1, 1, 1, 1, 1, 1, 12, 10, 7, 4, 2, 2, 1, 1, 1, 1, 1, 1, 13, 11, 8, 4, 2, 2
Offset: 0

Views

Author

Omar E. Pol, Jun 30 2012

Keywords

Comments

Column k contains k plateaus whose levels are the first k terms of A000124, therefore A000124(i) is the level of the i-th plateau of the column k when k -> infinity.
Column k contains the integers s>=1 repeated f(s) times, sorted, where f(s)=1 if s is not in A000124, otherwise, if A000124(c)=s, repeated f(s)=max(1,k+1-c) times. - R. J. Mathar, Jul 22 2012
It appears that this array can be represented by a structure in which the number of relevant nodes give A000005 (see also A210959). - Omar E. Pol, Jul 24 2012

Examples

			Illustration of initial terms of the 4th column:
------------------------------------------------------
Level    Graphic
------------------------------------------------------
10                                              *
9                                             *
8                                           *
7                                       * *
6                                     *
5                                   *
4                             * * *
3                           *
2                   * * * *
1         * * * * *
0
-------------------------------------------------------
Column 4: 1,1,1,1,1,2,2,2,2,3,4,4,4,5,6,7,7,8,9,10,...
-------------------------------------------------------
Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
2, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
3, 2, 1, 1, 1, 1, 1, 1, 1, 1,...
4, 3, 2, 1, 1, 1, 1, 1, 1, 1,...
5, 4, 2, 2, 1, 1, 1, 1, 1, 1,...
6, 5, 3, 2, 2, 1, 1, 1, 1, 1,...
7, 6, 4, 2, 2, 2, 1, 1, 1, 1,...
8, 7, 5, 3, 2, 2, 2, 1, 1, 1,...
9, 8, 6, 4, 2, 2, 2, 2, 1, 1,...

Links

Crossrefs

Columns 0-1: A000027, A028310.
Cf. A000124, A195825, A210843, A210959, A211970.

Programs

  • Maple
    A000124i := proc(n)
    local j;
    for j from 0 do
        if A000124(j) = n then
            return j;
        elif A000124(j) > n then
            return -1 ;
        end if;
    end do:
end proc:
A210992 := proc(n,k)
    local f,r,a,c;
    f := k+1  ;
    a := 1 ;
    for r from 0 to n do
        if f > 0 then
            f := f-1;
        else
            a := a+1 ;
            c := A000124i(a) ;
            f := 0 ;
            if c >= 0 then
                f := max(0,k-c) ;
            end if;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 22 2012

A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2 + j^2).

Original entry on oeis.org

1, 1, -2, 1, -1, 0, 1, -1, -1, 0, 1, -1, 0, 0, 2, 1, -1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, 1, -1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, 1, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, -2, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2017

Keywords

Examples

			Square array begins:
   1,   1,   1,   1,   1,   1, ...
  -2,  -1,  -1,  -1,  -1,  -1, ...
   0,  -1,   0,   0,   0,   0, ...
   0,   0,  -1,   0,   0,   0, ...
   2,   0,   0,  -1,   0,   0, ...
   0,   1,   0,   0,  -1,   0, ...

Links

Crossrefs

Columns k=0..10 give A002448, A010815, A106459, A113429, A089802, A232714, A244525, A281814 (absolute values), A205988 (absolute values), A281815 (absolute values), A247133.
Cf. A000041, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A006950, A015128, A036820, A051682, A051624, A051865, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A211970.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Sum[(-1)^i x^(k i (i - 1)/2 + i^2), {i, -n, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Product[(1 - x^((k + 2) i)) (1 - x^((k + 2) i - 1)) (1 - x^((k + 2) i - k - 1)), {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(x^(2 + k) QPochhammer[1/x, x^(2 + k)] QPochhammer[x^(-1 - k), x^(2 + k)] QPochhammer[x^(2 + k), x^(2 + k)])/((-1 + x) (-1 + x^(1 + k))), {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

Formula

G.f. of column 0: Sum_{j=-inf..inf} (-1)^j*x^A000290(j) = Product_{i>=1} (1 + x^i)/(1 - x^i) (convolution inverse of A015128).
G.f. of column 1: Sum_{j=-inf..inf} (-1)^j*x^A000326(j) = Product_{i>=1} (1 - x^i) (convolution inverse of A000041).
G.f. of column 2: Sum_{j=-inf..inf} (-1)^j*x^A000384(j) = Product_{i>=1} (1 - x^(2*i))/(1 + x^(2*i-1)) (convolution inverse of A006950).
G.f. of column 3: Sum_{j=-inf..inf} (-1)^j*x^A000566(j) = Product_{i>=1} (1 - x^(5*i))*(1 - x^(5*i-1))*(1 - x^(5*i-4)) (convolution inverse of A036820).
G.f. of column 4: Sum_{j=-inf..inf} (-1)^j*x^A000567(j) = Product_{i>=1} (1 - x^(6*i))*(1 - x^(6*i-1))*(1 - x^(6*i-5)) (convolution inverse of A195848).
G.f. of column 5: Sum_{j=-inf..inf} (-1)^j*x^A001106(j) = Product_{i>=1} (1 - x^(7*i))*(1 - x^(7*i-1))*(1 - x^(7*i-6)) (convolution inverse of A195849).
G.f. of column 6: Sum_{j=-inf..inf} (-1)^j*x^A001107(j) = Product_{i>=1} (1 - x^(8*i))*(1 - x^(8*i-1))*(1 - x^(8*i-7)) (convolution inverse of A195850).
G.f. of column 7: Sum_{j=-inf..inf} (-1)^j*x^A051682(j) = Product_{i>=1} (1 - x^(9*i))*(1 - x^(9*i-1))*(1 - x^(9*i-8)) (convolution inverse of A195851).
G.f. of column 8: Sum_{j=-inf..inf} (-1)^j*x^A051624(j) = Product_{i>=1} (1 - x^(10*i))*(1 - x^(10*i-1))*(1 - x^(10*i-9)) (convolution inverse of A195852).
G.f. of column 9: Sum_{j=-inf..inf} (-1)^j*x^A051865(j) = Product_{i>=1} (1 - x^(11*i))*(1 - x^(11*i-1))*(1 - x^(11*i-10)) (convolution inverse of A196933).
G.f. of column k: Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2+j^2) = Product_{i>=1} (1 - x^((k+2)*i))*(1 - x^((k+2)*i-1))*(1 - x^((k+2)*i-k-1)).
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