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 1-5 of 5 results.

A306914 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1-x)^k+x^k).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 3, 2, 0, 1, 4, 6, 0, 0, 1, 5, 10, 9, -4, 0, 1, 6, 15, 20, 9, -8, 0, 1, 7, 21, 35, 34, 0, -8, 0, 1, 8, 28, 56, 70, 48, -27, 0, 0, 1, 9, 36, 84, 126, 125, 48, -81, 16, 0, 1, 10, 45, 120, 210, 252, 200, 0, -162, 32, 0
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2019

Keywords

Examples

			Square array begins:
   1,  1,    1,    1,   1,    1,    1,    1, ...
   0,  2,    3,    4,   5,    6,    7,    8, ...
   0,  2,    6,   10,  15,   21,   28,   36, ...
   0,  0,    9,   20,  35,   56,   84,  120, ...
   0, -4,    9,   34,  70,  126,  210,  330, ...
   0, -8,    0,   48, 125,  252,  462,  792, ...
   0, -8,  -27,   48, 200,  461,  924, 1716, ...
   0,  0,  -81,    0, 275,  780, 1715, 3432, ...
   0, 16, -162, -164, 275, 1209, 2989, 6434, ...

Links

Crossrefs

Columns 1-9 give A000007, A099087, A057083, A099589(n+3), A289389(n+4), A306940, (-1)^n * A049018(n), A306941, A306942.
Cf. A039912, A306913, A306915, A307039, A307079, A307394.

Programs

  • Mathematica
    A[n_, k_] := SeriesCoefficient[1/((1-x)^k + x^k), {x, 0, n}];
Table[A[n-k+1, k], {n, 0, 11}, {k, n+1, 1, -1}] // Flatten (* Jean-François Alcover, Mar 20 2019 *)

Formula

A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n+k-1,k*j+k-1).
A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i+k-1,k*j+k-1) * binomial(n-i+k-1,k*j+k-1). - Seiichi Manyama, Apr 07 2019

A306915 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1-x)^k-x^k).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 3, 4, 8, 1, 4, 6, 8, 16, 1, 5, 10, 11, 16, 32, 1, 6, 15, 20, 21, 32, 64, 1, 7, 21, 35, 36, 42, 64, 128, 1, 8, 28, 56, 70, 64, 85, 128, 256, 1, 9, 36, 84, 126, 127, 120, 171, 256, 512, 1, 10, 45, 120, 210, 252, 220, 240, 342, 512, 1024
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2019

Keywords

Examples

			Square array begins:
     1,   1,   1,   1,   1,    1,    1,    1, ...
     2,   2,   3,   4,   5,    6,    7,    8, ...
     4,   4,   6,  10,  15,   21,   28,   36, ...
     8,   8,  11,  20,  35,   56,   84,  120, ...
    16,  16,  21,  36,  70,  126,  210,  330, ...
    32,  32,  42,  64, 127,  252,  462,  792, ...
    64,  64,  85, 120, 220,  463,  924, 1716, ...
   128, 128, 171, 240, 385,  804, 1717, 3432, ...
   256, 256, 342, 496, 715, 1365, 3017, 6436, ...

Links

Crossrefs

Columns (1+2),3-9 give A000079, A024495(n+2), A000749(n+3), A049016, A192080, A049017, A290995(n+7), A306939.
Cf. A039912, A101508, A306846, A306913, A306914, A307047, A307078, A307393.

Programs

  • Mathematica
    A[n_, k_] := Sum[Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[A[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 25 2021 *)

Formula

A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+k-1,k*j+k-1).
A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i+k-1,k*j+k-1) * binomial(n-i+k-1,k*j+k-1). - Seiichi Manyama, Apr 07 2019

A307047 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1+x)^k-x^k).

Original entry on oeis.org

1, 1, 0, 1, -2, 0, 1, -3, 4, 0, 1, -4, 6, -8, 0, 1, -5, 10, -9, 16, 0, 1, -6, 15, -20, 9, -32, 0, 1, -7, 21, -35, 36, 0, 64, 0, 1, -8, 28, -56, 70, -64, -27, -128, 0, 1, -9, 36, -84, 126, -125, 120, 81, 256, 0, 1, -10, 45, -120, 210, -252, 200, -240, -162, -512, 0
Offset: 0

Views

Author

Seiichi Manyama, Mar 21 2019

Keywords

Examples

			Square array begins:
   1,    1,    1,    1,    1,    1,     1,     1, ...
   0,   -2,   -3,   -4,   -5,   -6,    -7,    -8, ...
   0,    4,    6,   10,   15,   21,    28,    36, ...
   0,   -8,   -9,  -20,  -35,  -56,   -84,  -120, ...
   0,   16,    9,   36,   70,  126,   210,   330, ...
   0,  -32,    0,  -64, -125, -252,  -462,  -792, ...
   0,   64,  -27,  120,  200,  463,   924,  1716, ...
   0, -128,   81, -240, -275, -804, -1715, -3432, ...
   0,  256, -162,  496,  275, 1365,  2989,  6436, ...

Links

Crossrefs

Columns 1-7 give A000007, A122803, A000748, (-1)^n * A000749(n+3), A000750, A006090, A049018.
Cf. A039912 (square array A(n,k), n >= 0, k >= 2), A306913, A306914, A306915.

Programs

  • Mathematica
    T[n_, k_] := (-1)^n * Sum[(-1)^(j * Mod[k, 2]) * Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

Formula

A(n,k) = (-1)^n * Sum_{j=0..floor(n/k)} (-1)^((k mod 2) * j) * binomial(n+k-1,k*j+k-1).

A306913 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1+x)^k+x^k).

Original entry on oeis.org

1, 1, -2, 1, -2, 4, 1, -3, 2, -8, 1, -4, 6, 0, 16, 1, -5, 10, -11, -4, -32, 1, -6, 15, -20, 21, 8, 64, 1, -7, 21, -35, 34, -42, -8, -128, 1, -8, 28, -56, 70, -48, 85, 0, 256, 1, -9, 36, -84, 126, -127, 48, -171, 16, -512, 1, -10, 45, -120, 210, -252, 220, 0, 342, -32, 1024
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2019

Keywords

Examples

			Square array begins:
      1,  1,    1,    1,    1,    1,     1,     1, ...
     -2, -2,   -3,   -4,   -5,   -6,    -7,    -8, ...
      4,  2,    6,   10,   15,   21,    28,    36, ...
     -8,  0,  -11,  -20,  -35,  -56,   -84,  -120, ...
     16, -4,   21,   34,   70,  126,   210,   330, ...
    -32,  8,  -42,  -48, -127, -252,  -462,  -792, ...
     64, -8,   85,   48,  220,  461,   924,  1716, ...
   -128,  0, -171,    0, -385, -780, -1717, -3432, ...
    256, 16,  342, -164,  715, 1209,  3017,  6434, ...

Links

Crossrefs

Columns 1-2 give A122803, A108520.
Cf. A039912, A306914, A306915.

Programs

  • Mathematica
    A[n_, k_] := (-1)^n * Sum[(-1)^(Mod[k+1, 2] * j) * Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[A[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 25 2021 *)

Formula

A(n,k) = (-1)^n * Sum_{j=0..floor(n/k)} (-1)^(((k+1) mod 2) * j) * binomial(n+k-1,k*j+k-1).

A126713 The triangle K referred to in A038200, read along rows.

Original entry on oeis.org

1, -1, 1, 1, -3, 1, -1, 7, -4, 1, 1, -15, 10, -5, 1, -1, 31, -19, 15, -6, 1, 1, -63, 28, -35, 21, -7, 1, -1, 127, -28, 71, -56, 28, -8, 1, 1, -255, 1, -135, 126, -84, 36, -9, 1, -1, 511, 80, 255, -251, 210, -120, 45, -10, 1, 1, -1023, -242, -495, 451, -462, 330, -165, 55, -11, 1, -1, 2047, 485, 991, -726, 925, -792, 495, -220
Offset: 0

Views

Author

R. J. Mathar, Feb 12 2007

Keywords

Comments

This means the description of A038200 is slightly incorrect and ought be: "Row sums of triangle K(m,n), inverse to a triangle obtained from A020921 after eliminating the leftmost column."

Examples

			If the leftmost column of the triangle in A020921 is deleted we get
1
1 1
2 3 1
2 5 4 1
4 10 10 5 1
2 11 19 15 6 1
6 21 35 35 21 7 1
4 22 52 69 56 28 8 1
6 33 83 126 126 84 36 9 1
The present triangle is the inverse of this, namely
1
-1 1
1 -3 1
-1 7 -4 1
1 -15 10 -5 1
-1 31 -19 15 -6 1
1 -63 28 -35 21 -7 1
-1 127 -28 71 -56 28 -8 1
with row sums 1,0,-1,3,-8,21,-54,134,-318,720 of A038200.

Crossrefs

Cf. A039912.

Programs

  • Maple
    A020921 := proc(n,k) option remember; local divs; if n <= 0 then 1; elif k > n then 0; else divs := numtheory[divisors](n); add(numtheory[mobius](op(i,divs))*binomial(n/op(i,divs),k),i=1..nops(divs)); fi; end: A020921t := proc(n,k) option remember; A020921(n+1,k+1); end: TriLInv := proc(nmax) local a,row,col; a := array(0..nmax,0..nmax); for row from 0 to nmax do for col from row+1 to nmax do a[row,col] := 0; od; od; for row from 0 to nmax do for col from row to 0 by -1 do if row <> col then a[row,col] := -add(a[row,c]*A020921t(c,col),c=col+1..row)/A020921t(col,col); else a[row,col] := (1-add(a[row,c]*A020921t(c,col),c=col+1..row))/A020921t(col,col); fi; od; od; RETURN(a); end: nmax := 12 : a := TriLInv(nmax) : for row from 0 to nmax do for col from 0 to row do printf("%d, ",a[row,col]); od; od:
  • Mathematica
    f[n_] := (1/(1+x))*Sum[x^(k-1)/((1+x)^k-y*x^k), {k, 1, n+1}]; t[0, 0] = 1; t[n_, k_] := SeriesCoefficient[f[n], {x, 0, n}, {y, 0, k}]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 13 2013, after Vladeta Jovovic *)

Formula

G.f.: (1/(1+x))*Sum(x^(k-1)/((1+x)^k-y*x^k),k=1..infinity). - Vladeta Jovovic, Feb 26 2008
Showing 1-5 of 5 results.

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