A306914 -id:A306914 - cp's OEIS frontend

A307039 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k+x^k).

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, -2, 0, 1, 1, 1, 0, -4, 0, 1, 1, 1, 1, -3, -4, 0, 1, 1, 1, 1, 0, -9, 0, 0, 1, 1, 1, 1, 1, -4, -18, 8, 0, 1, 1, 1, 1, 1, 0, -14, -27, 16, 0, 1, 1, 1, 1, 1, 1, -5, -34, -27, 16, 0, 1, 1, 1, 1, 1, 1, 0, -20, -68, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, -6, -55, -116, 81, -32, 0

Offset: 0

Seiichi Manyama, Mar 21 2019

			Square array begins:
   1,  1,   1,    1,    1,   1,   1,  1, ...
   0,  1,   1,    1,    1,   1,   1,  1, ...
   0,  0,   1,    1,    1,   1,   1,  1, ...
   0, -2,   0,    1,    1,   1,   1,  1, ...
   0, -4,  -3,    0,    1,   1,   1,  1, ...
   0,  0, -18,  -14,   -5,   0,   1,  1, ...
   0,  8, -27,  -34,  -20,  -6,   0,  1, ...
   0, 16, -27,  -68,  -55, -27,  -7,  0, ...
   0, 16,   0, -116, -125, -83, -35, -8, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-9 give A000007, A146559, A057681, A099586, A289306, A307040, A307041, A307044, A307045.

Cf. A306846, A306914.

T[n_, k_] := Sum[(-1)^j * Binomial[n, k*j], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 13}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

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

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

Seiichi Manyama, Mar 16 2019

			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, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

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.

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 *)

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

Seiichi Manyama, Mar 21 2019

			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, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

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.

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 *)

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

A307079 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k+x^k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 0, 1, 1, 2, 3, 3, -4, 1, 1, 2, 3, 4, 0, -8, 1, 1, 2, 3, 4, 4, -9, -8, 1, 1, 2, 3, 4, 5, 0, -27, 0, 1, 1, 2, 3, 4, 5, 5, -14, -54, 16, 1, 1, 2, 3, 4, 5, 6, 0, -48, -81, 32, 1, 1, 2, 3, 4, 5, 6, 6, -20, -116, -81, 32, 1

Offset: 0

Seiichi Manyama, Mar 22 2019

			Square array begins:
   1,  1,   1,    1,   1,   1, 1, 1, 1, ...
   1,  2,   2,    2,   2,   2, 2, 2, 2, ...
   1,  2,   3,    3,   3,   3, 3, 3, 3, ...
   1,  0,   3,    4,   4,   4, 4, 4, 4, ...
   1, -4,   0,    4,   5,   5, 5, 5, 5, ...
   1, -8,  -9,    0,   5,   6, 6, 6, 6, ...
   1, -8, -27,  -14,   0,   6, 7, 7, 7, ...
   1,  0, -54,  -48, -20,   0, 7, 8, 8, ...
   1, 16, -81, -116, -75, -27, 0, 8, 9, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-6 give A000012, A099087, A057682(n+1), A099587(n+1), A289321(n+1), A307089.

Cf. A306914, A307039, A307078.

T[n_, k_] := Sum[(-1)^j * Binomial[n+1, k*j+1], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

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

A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i,k*j) * binomial(n-i,k*j).

A307394 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-4))/((1-x)^k+x^k).

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 4, 9, 10, 1, 4, 10, 14, 15, 1, 4, 10, 19, 15, 21, 1, 4, 10, 20, 28, 8, 28, 1, 4, 10, 20, 34, 28, -7, 36, 1, 4, 10, 20, 35, 48, 1, -22, 45, 1, 4, 10, 20, 35, 55, 48, -80, -21, 55, 1, 4, 10, 20, 35, 56, 75, 0, -242, 12, 66, 1, 4, 10, 20, 35, 56, 83, 75, -164, -485, 77, 78

Offset: 0

Seiichi Manyama, Apr 07 2019

			Square array begins:
    1,   1,    1,    1,  1,   1,   1,   1,   1, ...
    3,   4,    4,    4,  4,   4,   4,   4,   4, ...
    6,   9,   10,   10, 10,  10,  10,  10,  10, ...
   10,  14,   19,   20, 20,  20,  20,  20,  20, ...
   15,  15,   28,   34, 35,  35,  35,  35,  35, ...
   21,   8,   28,   48, 55,  56,  56,  56,  56, ...
   28,  -7,    1,   48, 75,  83,  84,  84,  84, ...
   36, -22,  -80,    0, 75, 110, 119, 120, 120, ...
   45, -21, -242, -164,  0, 110, 154, 164, 165, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-5 give A000217(n+1), A279230, A307395, A099589(n+3), A289388(n+3).

Cf. A306914, A307039, A307079, A307393.

T[n_, k_] := Sum[(-1)^j * Binomial[n+3, k*j + 3], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n+3,k*j+3).

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

Seiichi Manyama, Mar 16 2019

			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, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-2 give A122803, A108520.

Cf. A039912, A306914, A306915.

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 *)

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).

A306940 Expansion of 1/((1 - x)^6 + x^6).

Original entry on oeis.org

1, 6, 21, 56, 126, 252, 461, 780, 1209, 1638, 1638, 0, -6187, -23238, -63783, -151316, -326382, -652764, -1217483, -2107560, -3322995, -4538430, -4538430, 0, 16942381, 63239286, 172791861, 408855776, 880983606, 1761967212, 3287837741, 5694626340

Offset: 0

Seiichi Manyama, Mar 17 2019

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-2).

Column 6 of A306914.

CoefficientList[Series[1/((1 - x)^6 + x^6), {x, 0, 31}], x] (* Amiram Eldar, May 25 2021 *)
LinearRecurrence[{6,-15,20,-15,6,-2},{1,6,21,56,126,252},40] (* Harvey P. Dale, May 31 2021 *)

{a(n) = sum(k=0, n\6, (-1)^k*binomial(n+5, 6*k+5))}

N=66; x='x+O('x^N); Vec(1/((1-x)^6+x^6))

a(n) = Sum_{k=0..floor(n/6)} (-1)^k*binomial(n+5,6*k+5).

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - 2*a(n-6) for n > 5.

A306941 Expansion of 1/((1 - x)^8 + x^8).

Original entry on oeis.org

1, 8, 36, 120, 330, 792, 1716, 3432, 6434, 11424, 19312, 31008, 46512, 62016, 62016, 0, -245156, -961376, -2787760, -7065760, -16478280, -36159840, -75522960, -151045920, -290021896, -534308096, -940325760, -1564496128, -2406819200, -3249142272, -3249142272

Offset: 0

Seiichi Manyama, Mar 17 2019

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-2).

Column 8 of A306914.

CoefficientList[Series[1/((1 - x)^8 + x^8), {x, 0, 30}], x] (* Amiram Eldar, May 25 2021 *)

{a(n) = sum(k=0, n\8, (-1)^k*binomial(n+7, 8*k+7))}

N=66; x='x+O('x^N); Vec(1/((1-x)^8+x^8))

a(n) = Sum_{k=0..floor(n/8)} (-1)^k*binomial(n+7,8*k+7).

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - 2*a(n-8) for n > 7.

A306942 Expansion of 1/((1 - x)^9 + x^9).

Original entry on oeis.org

1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24309, 43740, 75411, 124830, 197505, 293436, 389367, 389367, 0, -1562274, -6216183, -18365697, -47600136, -113879520, -257123889, -554298228, -1148646906, -2297293812, -4443424371, -8313049440, -15011769204

Offset: 0

Seiichi Manyama, Mar 17 2019

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9).

Column 9 of A306914.

Cf. A306939.

CoefficientList[Series[1/((1-x)^9+x^9),{x,0,30}],x] (* Harvey P. Dale, Sep 25 2019 *)

{a(n) = sum(k=0, n\9, (-1)^k*binomial(n+8, 9*k+8))}

N=66; x='x+O('x^N); Vec(1/((1-x)^9+x^9))

a(n) = Sum_{k=0..floor(n/9)} (-1)^k*binomial(n+8,9*k+8).

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) for n > 7.

A307668 A(n,k) = Sum_{j=0..floor(n/k)} (-1)^jbinomial(2n,k*j+n), square array A(n,k) read by antidiagonals, for n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 2, 5, 10, 1, 2, 6, 14, 35, 1, 2, 6, 19, 43, 126, 1, 2, 6, 20, 62, 142, 462, 1, 2, 6, 20, 69, 207, 494, 1716, 1, 2, 6, 20, 70, 242, 705, 1780, 6435, 1, 2, 6, 20, 70, 251, 858, 2445, 6563, 24310, 1, 2, 6, 20, 70, 252, 912, 3068, 8622, 24566, 92378

Offset: 0

Seiichi Manyama, Apr 20 2019

			Square array begins:
      1,    1,    1,     1,     1,     1,     1, ...
      1,    2,    2,     2,     2,     2,     2, ...
      3,    5,    6,     6,     6,     6,     6, ...
     10,   14,   19,    20,    20,    20,    20, ...
     35,   43,   62,    69,    70,    70,    70, ...
    126,  142,  207,   242,   251,   252,   252, ...
    462,  494,  705,   858,   912,   923,   924, ...
   1716, 1780, 2445,  3068,  3341,  3418,  3431, ...
   6435, 6563, 8622, 11051, 12310, 12750, 12854, ...

Columns 1-2 give A088218, A005317.

Cf. A306914, A307039, A307394, A307665.

T[n_, k_] := Sum[(-1)^j*Binomial[2*n, k*j + n], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 13 2021*)

cp's OEIS Frontend

A307039 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k+x^k).

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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).

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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).

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A307079 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k+x^k).

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A307394 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-4))/((1-x)^k+x^k).

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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).

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A306940 Expansion of 1/((1 - x)^6 + x^6).

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A306941 Expansion of 1/((1 - x)^8 + x^8).

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A307668 A(n,k) = Sum_{j=0..floor(n/k)} (-1)^jbinomial(2n,k*j+n), square array A(n,k) read by antidiagonals, for n >= 0, k >= 1.