A306846 -id:A306846 - cp's OEIS frontend

A167009 a(n) = Sum_{k=0..n} C(n^2, n*k).

1, 2, 8, 170, 16512, 6643782, 11582386286, 79450506979090, 2334899414608412672, 265166261617029717011822, 128442558588779813655233443038, 238431997806538515396060130910954852

Offset: 0

Paul D. Hanna, Nov 17 2009

nonn

			The triangle A209330 of coefficients C(n^2, n*k), n>=k>=0, begins:
  1;
  1,       1;
  1,       6,          1;
  1,      84,         84,          1;
  1,    1820,      12870,       1820,          1;
  1,   53130,    3268760,    3268760,      53130,       1;
  1, 1947792, 1251677700, 9075135300, 1251677700, 1947792,     1; ...
in which the row sums form this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 0..58
Vaclav Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013.

Cf. A014062, A167006, A167010, A209330, A218792, A306846.

[(&+[Binomial(n^2, n*j): j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022

Table[Sum[Binomial[n^2,n*k],{k,0,n}],{n,0,15}] (* Harvey P. Dale, Dec 11 2011 *)

PARI
```
a(n)=sum(k=0,n,binomial(n^2,n*k))
```

[sum(binomial(n^2, n*j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Ignoring initial term, equals the logarithmic derivative of A167006. - Paul D. Hanna, Nov 18 2009
If n is even then a(n) ~ c * 2^(n^2 + 1/2)/(n*sqrt(Pi)), where c = Sum_{k = -infinity..infinity} exp(-2*k^2) = 1.271341522189... (see A218792). - Vaclav Kotesovec, Nov 05 2012
If n is odd then c = Sum_{k = -infinity..infinity} exp(-2*(k+1/2)^2) = 1.23528676585389... - Vaclav Kotesovec, Nov 06 2012
a(n) = A306846(n^2,n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n - x^n) for n > 0. - Seiichi Manyama, Oct 11 2021

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).

Original entry on oeis.org

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

A306847 a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 463, 926, 1730, 3095, 5461, 9829, 18565, 37130, 77540, 164921, 349525, 728575, 1486675, 2973350, 5858126, 11450531, 22369621, 43942081, 87087001, 174174002, 350739488, 708653429, 1431655765, 2884834891, 5791193143

Offset: 0

Seiichi Manyama, Mar 13 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).

Column 6 of A306846.

a[n_] := Sum[Binomial[n, 6*k], {k, 0, Floor[n/6]}]; Array[a, 36, 0] (* Amiram Eldar, Jun 21 2021 *)

{a(n) = sum(k=0, n\6, 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) for n > 5.
a(n) = (4^n + (1 - t)^n + (1 + t)^n + (3 - t)^n + (3 + t)^n)/(6*2^n) for n > 0 and a(0) = 1, where t = i*sqrt(3) and i = sqrt(-1). - Bruno Berselli, Mar 13 2019

A306860 a(n) = Sum_{k=0..floor(n/9)} binomial(n,9*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 11, 56, 221, 716, 2003, 5006, 11441, 24311, 48622, 92398, 168151, 295261, 504736, 850840, 1442101, 2523676, 4686826, 9373652, 20030039, 44612702, 100804436, 226444616, 499685777, 1076832989, 2261792303, 4631710931, 9263421862

Offset: 0

Seiichi Manyama, Mar 14 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,-9,2).

Column 9 of A306846.

a[n_] := Sum[Binomial[n, 9*k], {k, 0, Floor[n/9]}]; Array[a, 40, 0] (* Amiram Eldar, Jun 13 2021 *)

{a(n) = sum(k=0, n\9, 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) + 2*a(n-9) for n > 8.

A307078 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, 3, 1, 2, 7, 1, 2, 4, 15, 1, 2, 3, 8, 31, 1, 2, 3, 5, 16, 63, 1, 2, 3, 4, 10, 32, 127, 1, 2, 3, 4, 6, 21, 64, 255, 1, 2, 3, 4, 5, 12, 43, 128, 511, 1, 2, 3, 4, 5, 7, 28, 86, 256, 1023, 1, 2, 3, 4, 5, 6, 14, 64, 171, 512, 2047, 1, 2, 3, 4, 5, 6, 8, 36, 136, 341, 1024, 4095

Offset: 0

Seiichi Manyama, Mar 22 2019

			Square array begins:
     1,   1,   1,   1,  1,  1,  1,  1, 1, ...
     3,   2,   2,   2,  2,  2,  2,  2, 2, ...
     7,   4,   3,   3,  3,  3,  3,  3, 3, ...
    15,   8,   5,   4,  4,  4,  4,  4, 4, ...
    31,  16,  10,   6,  5,  5,  5,  5, 5, ...
    63,  32,  21,  12,  7,  6,  6,  6, 6, ...
   127,  64,  43,  28, 14,  8,  7,  7, 7, ...
   255, 128,  86,  64, 36, 16,  9,  8, 8, ...
   511, 256, 171, 136, 93, 45, 18, 10, 9, ...

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

Columns 1-6 give A126646, A000079, A024494(n+1), A038504(n+1), A133476(n+1), A119336.

Cf. A306846, A306915, A307079.

T[n_, k_] := Sum[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)} binomial(n+1,k*j+1).

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

A307393 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, 5, 1, 4, 16, 1, 4, 11, 42, 1, 4, 10, 26, 99, 1, 4, 10, 21, 57, 219, 1, 4, 10, 20, 42, 120, 466, 1, 4, 10, 20, 36, 84, 247, 968, 1, 4, 10, 20, 35, 64, 169, 502, 1981, 1, 4, 10, 20, 35, 57, 120, 340, 1013, 4017, 1, 4, 10, 20, 35, 56, 93, 240, 682, 2036, 8100

Offset: 0

Seiichi Manyama, Apr 07 2019

			Square array begins:
     1,   1,   1,   1,   1,   1,   1,   1, ...
     5,   4,   4,   4,   4,   4,   4,   4, ...
    16,  11,  10,  10,  10,  10,  10,  10, ...
    42,  26,  21,  20,  20,  20,  20,  20, ...
    99,  57,  42,  36,  35,  35,  35,  35, ...
   219, 120,  84,  64,  57,  56,  56,  56, ...
   466, 247, 169, 120,  93,  85,  84,  84, ...
   968, 502, 340, 240, 165, 130, 121, 120, ...

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

Columns 1-5 give A002662(n+3), A125128(n+1), A111927(n+3), A000749(n+3), A139748(n+3).

Cf. A306915, A306846, A307078, A307394.

T[n_, k_] := Sum[Binomial[n+3, k*j + 3], {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) = Sum_{j=0..floor(n/k)} binomial(n+3,k*j+3).

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

A306852 a(n) = Sum_{k=0..floor(n/7)} binomial(n,7*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1717, 3434, 6451, 11561, 20129, 34885, 62017, 116281, 232562, 490337, 1062601, 2309385, 4950751, 10381281, 21242341, 42484682, 83411715, 161766061, 312168761, 603861897, 1178135905, 2326683921, 4653367842

Offset: 0

Seiichi Manyama, Mar 14 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,-7,2).

Column 7 of A306846.

a[n_] := Sum[Binomial[n,7*k], {k,0,Floor[n/7]}]; Array[a, 36, 0] (* Amiram Eldar, May 25 2021 *)

{a(n) = sum(k=0, n\7, 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) + 2*a(n-7) for n > 6.

A306859 a(n) = Sum_{k=0..floor(n/8)} binomial(n,8*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 10, 46, 166, 496, 1288, 3004, 6436, 12872, 24328, 43912, 76552, 130816, 223840, 394384, 735472, 1470944, 3124576, 6874336, 15260896, 33550336, 72274816, 151869376, 311058496, 622116992, 1219254400, 2353246336, 4500697216, 8589869056

Offset: 0

Seiichi Manyama, Mar 14 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).

Column 8 of A306846.

a[n_] := Sum[Binomial[n,8*k], {k,0,Floor[n/8]}]; Array[a, 37, 0] (* Amiram Eldar, May 25 2021 *)

{a(n) = sum(k=0, n\8, 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) for n > 7.

A307665 A(n,k) = Sum_{j=0..floor(n/k)} binomial(2n,kj+n), square array A(n,k) read by antidiagonals, for n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 3, 1, 2, 11, 1, 2, 7, 42, 1, 2, 6, 26, 163, 1, 2, 6, 21, 99, 638, 1, 2, 6, 20, 78, 382, 2510, 1, 2, 6, 20, 71, 297, 1486, 9908, 1, 2, 6, 20, 70, 262, 1145, 5812, 39203, 1, 2, 6, 20, 70, 253, 990, 4447, 22819, 155382, 1, 2, 6, 20, 70, 252, 936, 3796, 17358, 89846, 616666

Offset: 0

Seiichi Manyama, Apr 20 2019

			Square array begins:
      1,    1,    1,    1,    1,    1,    1,    1, ...
      3,    2,    2,    2,    2,    2,    2,    2, ...
     11,    7,    6,    6,    6,    6,    6,    6, ...
     42,   26,   21,   20,   20,   20,   20,   20, ...
    163,   99,   78,   71,   70,   70,   70,   70, ...
    638,  382,  297,  262,  253,  252,  252,  252, ...
   2510, 1486, 1145,  990,  936,  925,  924,  924, ...
   9908, 5812, 4447, 3796, 3523, 3446, 3433, 3432, ...

Columns 1-2 give A032443, A114121.

Cf. A306846, A306915, A307393, A307668.

T[n_, k_] := Sum[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

A167009 a(n) = Sum_{k=0..n} C(n^2, n*k).

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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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A306847 a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k).

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A306860 a(n) = Sum_{k=0..floor(n/9)} binomial(n,9*k).

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A307078 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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A307393 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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A306852 a(n) = Sum_{k=0..floor(n/7)} binomial(n,7*k).

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A307665 A(n,k) = Sum_{j=0..floor(n/k)} binomial(2n,kj+n), square array A(n,k) read by antidiagonals, for n >= 0, k >= 1.