A307039 -id:A307039 - cp's OEIS frontend

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).

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

Seiichi Manyama, Mar 16 2019

			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, ...

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

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.

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 *)

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

A289306 a(n) = Sum_{k >= 0}(-1)^kbinomial(n,5k).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, -5, -20, -55, -125, -250, -450, -725, -1000, -1000, 0, 3625, 13125, 34375, 76875, 153750, 278125, 450000, 621875, 621875, 0, -2250000, -8140625, -21312500, -47656250, -95312500, -172421875, -278984375, -385546875, -385546875, 0, 1394921875

Offset: 0

Vladimir Shevelev, Jul 02 2017

{A289306, A289321, A289387, A289388, A289389} is the difference analog of the trigonometric functions {k_1(x), k_2(x), k_3(x), k_4(x), k_5(x)} of order 5. For the definitions of {k_i(x)} and the difference analog {K_i (n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Jul 24 2017

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

Seiichi Manyama, Table of n, a(n) for n = 0..3000
John B. Dobson, A matrix variation on Ramus's identity for lacunary sums of binomial coefficients, arXiv preprint arXiv:1610.09361 [math.NT], 2016.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5).

Column 5 of A307039.

Cf. A139398, A133476, A139714, A139748, A139761.

Table[Sum[(-1)^k*Binomial[n, 5 k], {k, 0, n}], {n, 0, 36}] (* or *)
CoefficientList[Series[-((-1 + x)^4/((-1 + x)^5 - x^5)), {x, 0, 36}], x] (* Michael De Vlieger, Jul 04 2017 *)
LinearRecurrence[{5,-10,10,-5},{1,1,1,1,1},40] (* Harvey P. Dale, Dec 23 2018 *)

a(n) = sum(k=0, n\5, (-1)^k*binomial(n,5*k)); \\ Michel Marcus, Jul 02 2017

G.f.: -((-1+x)^4/((-1+x)^5-x^5)). - Peter J. C. Moses, Jul 02 2017
For n>=1, a(n) = (2/5)*(phi+2)^(n/2)*(cos(Pi*n/10) + (phi-1)^n*cos(3 * Pi* n/10)), where phi is the golden ratio. In particular, a(n) = 0 if and only if n==5 (mod 10).
a(n+m) = a(n)*a(m) - K_5(n)*K_2(m) - K_4(n)*K_3(m) - K_3(n)*K_4(m) - K_2(n)*K_5(m), where K_2 is A289321, K_3 is A289387, K_4 is A289388, K_5 is A289389. - Vladimir Shevelev, Jul 24 2017

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).

A307040 a(n) = Sum_{k=0..floor(n/6)} (-1)^kbinomial(n,6k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, -6, -27, -83, -209, -461, -922, -1702, -2911, -4549, -6187, -6187, 0, 23238, 87021, 238337, 564719, 1217483, 2434966, 4542526, 7865521, 12403951, 16942381, 16942381, 0, -63239286, -236031147, -644886923, -1525870529, -3287837741, -6575675482

Offset: 0

Seiichi Manyama, Mar 21 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 A307039.

Cf. A306847.

a[n_] := Sum[(-1)^k * Binomial[n,6*k], {k,0,Floor[n/6]}]; Array[a, 37, 0] (* Amiram Eldar, May 25 2021 *)

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

N=66; x='x+O('x^N); Vec((1-x)^5/((1-x)^6+x^6))

G.f.: (1 - x)^5/((1 - x)^6 + x^6).

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.

A307041 a(n) = Sum_{k=0..floor(n/7)} (-1)^kbinomial(n,7k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 0, -7, -35, -119, -329, -791, -1715, -3430, -6419, -11319, -18767, -28763, -38759, -38759, 0, 149205, 571781, 1613129, 3964051, 8934121, 18874261, 37748522, 71705865, 129080161, 218205281, 339081225, 459957169, 459957169, 0, -1749692735

Offset: 0

Seiichi Manyama, Mar 21 2019

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7).

Column 7 of A307039.

Cf. A306852.

a[n_] := Sum[(-1)^k * Binomial[n,7*k], {k,0,Floor[n/7]}]; Array[a, 37, 0] (* Amiram Eldar, May 25 2021 *)

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

N=66; x='x+O('x^N); Vec((1-x)^6/((1-x)^7+x^7))

G.f.: (1 - x)^6/((1 - x)^7 + x^7).

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) for n > 6.

A307044 a(n) = Sum_{k=0..floor(n/8)} (-1)^kbinomial(n,8k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, -8, -44, -164, -494, -1286, -3002, -6434, -12868, -24292, -43604, -74612, -121124, -183140, -245156, -245156, 0, 961376, 3749136, 10814896, 27293176, 63453016, 138975976, 290021896, 580043792, 1114351888, 2054677648, 3619173776

Offset: 0

Seiichi Manyama, Mar 21 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 A307039.

Cf. A306859.

a[n_] := Sum[(-1)^k * Binomial[n,8*k], {k,0,Floor[n/8]}]; Array[a, 36, 0] (* Amiram Eldar, May 25 2021 *)

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

N=66; x='x+O('x^N); Vec((1-x)^7/((1-x)^8+x^8))

G.f.: (1 - x)^7/((1 - x)^8 + x^8).

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.

A307045 a(n) = Sum_{k=0..floor(n/9)} (-1)^kbinomial(n,9k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 0, -9, -54, -219, -714, -2001, -5004, -11439, -24309, -48618, -92358, -167769, -292599, -490104, -783540, -1172907, -1562274, -1562274, 0, 6216183, 24581880, 72182016, 186061536, 443185425, 997483653, 2146130559, 4443424371, 8886848742

Offset: 0

Seiichi Manyama, Mar 21 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 A307039.

Cf. A306860.

a[n_] := Sum[(-1)^k * Binomial[n,9*k], {k,0,Floor[n/9]}]; Array[a, 37, 0] (* Amiram Eldar, May 25 2021 *)

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

N=66; x='x+O('x^N); Vec((1-x)^8/((1-x)^9+x^9))

G.f.: (1 - x)^8/((1 - x)^9 + x^9).

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 > 8.

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

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).

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A289306 a(n) = Sum_{k >= 0}(-1)^k*binomial(n,5*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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A307040 a(n) = Sum_{k=0..floor(n/6)} (-1)^k*binomial(n,6*k).

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A307041 a(n) = Sum_{k=0..floor(n/7)} (-1)^k*binomial(n,7*k).

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A307044 a(n) = Sum_{k=0..floor(n/8)} (-1)^k*binomial(n,8*k).

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PARI

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A307045 a(n) = Sum_{k=0..floor(n/9)} (-1)^k*binomial(n,9*k).

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A289306 a(n) = Sum_{k >= 0}(-1)^kbinomial(n,5k).

A307040 a(n) = Sum_{k=0..floor(n/6)} (-1)^kbinomial(n,6k).

A307041 a(n) = Sum_{k=0..floor(n/7)} (-1)^kbinomial(n,7k).

A307044 a(n) = Sum_{k=0..floor(n/8)} (-1)^kbinomial(n,8k).

A307045 a(n) = Sum_{k=0..floor(n/9)} (-1)^kbinomial(n,9k).

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.