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.

Previous Showing 31-38 of 38 results.

A210764 Square array T(n,k), n>=0, k>=0, read by antidiagonals in which column k gives the partial sums of column k of A144064.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 8, 4, 1, 1, 12, 18, 13, 5, 1, 1, 19, 38, 35, 19, 6, 1, 1, 30, 74, 86, 59, 26, 7, 1, 1, 45, 139, 194, 164, 91, 34, 8, 1, 1, 67, 249, 415, 416, 281, 132, 43, 9, 1, 1, 97, 434, 844, 990, 787, 447, 183, 53, 10, 1
Offset: 0

Views

Author

Omar E. Pol, Jun 27 2012

Keywords

Comments

It appears that row 2 is A034856.
Observation:
Column 1 is the EULER transform of 2,1,1,1,1,1,1,1...
Column 2 is the EULER transform of 3,2,2,2,2,2,2,2...

Examples

			Array begins:
1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
1,   4,   8,  13,  19,  26,  34,  43,  53,
1,   7,  18,  35,  59,  91, 132, 183,
1,  12,  38,  86, 164, 281, 447,
1,  19,  74, 194, 416, 787,
1,  30, 139, 415, 990,
1,  45, 249, 844,
1,  67, 434,
1,  97,
1,

Links

Crossrefs

Columns (0-3): A000012, A000070, A000713, A210843.
Rows (0-1): A000012, A000027.
Main diagonal gives A303070.
Cf. A000007, A000041, A005758, A006922, A000712, A000716, A023003-A023021, A144064, A195825, A211970.

Programs

  • Maple
    with(numtheory):
etr:= proc(p) local b;
        b:= proc(n) option remember; `if`(n=0, 1,
              add(add(d*p(d), d=divisors(j))*b(n-j), j=1..n)/n)
            end
      end:
A:= (n, k)-> etr(j-> k +`if`(j=1, 1, 0))(n):
seq(seq(A(d-k, k), k=0..d), d=0..14); # Alois P. Heinz, May 20 2013
  • Mathematica
    etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[Function[{j}, k + If[j == 1, 1, 0]]][n]; Table[Table[A[d-k, k], {k, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

A233758 Bisection of A006950 (the even part).

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 28, 43, 70, 105, 161, 236, 350, 501, 722, 1016, 1431, 1981, 2741, 3740, 5096, 6868, 9233, 12306, 16357, 21581, 28394, 37128, 48406, 62777, 81182, 104494, 134131, 171467, 218607, 277691, 351841, 444314, 559727, 703002, 880896, 1100775
Offset: 1

Views

Author

Omar E. Pol, Jan 11 2014

Keywords

Comments

See Zaletel-Mong paper, page 14, FIG. 11: C2a is this sequence, C2b is A233759, C2c is A015128.

Links

Crossrefs

Cf. A000009, A015128, A006950, A069910, A069911, A195825, A195836, A211970, A233759.

Programs

  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i - Mod[i, 2]]]]];
a[n_] := b[2 n - 2, 2 n - 2];
Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz in A006950 *)

A233759 Bisection of A006950 (the odd part).

Original entry on oeis.org

1, 2, 4, 7, 13, 21, 35, 55, 86, 130, 196, 287, 420, 602, 858, 1206, 1687, 2331, 3206, 4368, 5922, 7967, 10670, 14193, 18803, 24766, 32490, 42411, 55159, 71416, 92152, 118434, 151725, 193676, 246491, 312677, 395537, 498852, 627509, 787171, 985043, 1229494
Offset: 1

Views

Author

Omar E. Pol, Jan 11 2014

Keywords

Comments

See Zaletel-Mong paper, page 14, FIG. 11: C2a is A233758, C2b is this sequence, C2c is A015128.

Links

Crossrefs

Cf. A000009, A015128, A006950, A069910, A069911, A195825, A195836, A211970, A233758.

Programs

  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i - Mod[i, 2]]]]];
a[n_] := b[2 n - 1, 2 n - 1];
Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz in A006950 *)

A249120 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers of A210843 multiplied by A000330(k), and the first element of column k is in row A000217(k).

Original entry on oeis.org

1, 4, 13, 5, 35, 20, 86, 65, 194, 175, 14, 415, 430, 56, 844, 970, 182, 1654, 2075, 490, 3133, 4220, 1204, 30, 5773, 8270, 2716, 120, 10372, 15665, 5810, 390, 18240, 28865, 11816, 1050, 31449, 51860, 23156, 2580, 53292, 91200, 43862, 5820, 55, 88873, 157245, 80822, 12450, 220, 146095, 266460, 145208, 25320, 715
Offset: 1

Views

Author

Omar E. Pol, Dec 14 2014

Keywords

Comments

Conjecture: gives an identity for the sum of all divisors of all positive integers <= n. Alternating sum of row n equals A024916(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A024916(n).
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
Column 1 is A210843.
Column k lists the partial sums of the k-th column of triangle A252117 which gives an identity for sigma.
The first element of column k is A000330(k).
The second element of column k is A002492(k).

Examples

			Triangle begins:
       1;
       4;
      13,       5;
      35,      20;
      86,      65;
     194,     175,      14;
     415,     430,      56;
     844,     970,     182;
    1654,    2075,     490;
    3133,    4220,    1204,     30;
    5773,    8270,    2716,    120;
   10372,   15665,    5810,    390;
   18240,   28865,   11816,   1050;
   31449,   51860,   23156,   2580;
   53292,   91200,   43862,   5820,    55;
   88873,  157245,   80822,  12450,   220;
  146095,  266460,  145208,  25320,   715;
  236977,  444365,  255360,  49620,  1925;
  379746,  730475,  440286,  93990,  4730;
  601656, 1184885,  746088, 173190, 10670;
  943305, 1898730, 1244222, 311160, 22825,   91;
  ...
For n = 6 the sum of all divisors of all positive integers <= 6 is [1] + [1+2] + [1+3] + [1+2+4] + [1+5] + [1+2+3+6] = 1 + 3 + 4 + 7 + 6 + 12 = 33. On the other hand the 6th row of triangle is 194, 175, 14, so the alternating row sum is 194 - 175 + 14 = 33, equaling the sum of all divisors of all positive integers <= 6.

Crossrefs

Cf. A000203, A000217, A000330, A002492, A003056, A024916, A195825, A196020, A210843, A211970, A236104, A252117.

A384999 Irregular triangle read by rows: T(n,k) is the total number of partitions of all numbers <= n with k designated summands, n >= 0, 0 <= k <= A003056(n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 8, 1, 1, 15, 4, 1, 21, 13, 1, 33, 28, 1, 1, 41, 58, 4, 1, 56, 103, 13, 1, 69, 170, 35, 1, 87, 269, 77, 1, 1, 99, 404, 158, 4, 1, 127, 579, 298, 13, 1, 141, 810, 529, 35, 1, 165, 1116, 880, 86, 1, 189, 1470, 1431, 183, 1, 1, 220, 1935, 2214, 371, 4, 1, 238, 2475, 3348, 701, 13
Offset: 0

Views

Author

Omar E. Pol, Jul 22 2025

Keywords

Comments

When part i has multiplicity j > 0 exactly one part i is "designated".
The length of the row n is A002024(n+1) = 1 + A003056(n), hence the first positive element in column k is in the row A000217(k).
Column k gives the partial sums of the column k of A385001.
Columns converge to A210843 which is also the partial sums of A000716.

Examples

			Triangle begins:
---------------------------------------------
   n\k:   0    1     2      3     4    5   6
---------------------------------------------
   0 |    1;
   1 |    1,   1;
   2 |    1,   4;
   3 |    1,   8,    1;
   4 |    1,  15,    4;
   5 |    1,  21,   13;
   6 |    1,  33,   28,     1;
   7 |    1,  41,   58,     4;
   8 |    1,  56,  103,    13;
   9 |    1,  69,  170,    35;
  10 |    1,  87,  269,    77,    1;
  11 |    1,  99,  404,   158,    4;
  12 |    1, 127,  579,   298,   13;
  13 |    1, 141,  810,   529,   35;
  14 |    1, 165, 1116,   880,   86;
  15 |    1, 189, 1470,  1431,  183,   1;
  16 |    1, 220, 1935,  2214,  371,   4;
  17 |    1, 238, 2475,  3348,  701,  13;
  18 |    1, 277, 3156,  4894, 1269,  35;
  19 |    1, 297, 3921,  7036, 2187,  86;
  20 |    1, 339, 4866,  9871, 3639, 194;
  21 |    1, 371, 5906, 13629, 5872, 402,  1;
  ...

Links

Crossrefs

Columns: A000012 (k=0), A024916 (k=1).
Cf. A000217, A000716, A002024, A003056, A077285, A195825, A210843, A211970, A385001.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(expand(b(n-i*j, i-1)*j*x), j=1..n/i)))
    end:
g:= proc(n) option remember; `if`(n<0, 0, g(n-1)+b(n$2)) end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 22 2025

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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