A287316 -id:A287316 - cp's OEIS frontend

A287314 Triangle read by rows, the coefficients of the polynomials generating the columns of A287316.

1, 0, 1, 0, -1, 2, 0, 4, -9, 6, 0, -33, 82, -72, 24, 0, 456, -1225, 1250, -600, 120, 0, -9460, 27041, -30600, 17700, -5400, 720, 0, 274800, -826336, 1011017, -661500, 249900, -52920, 5040, 0, -10643745, 33391954, -43471624, 31149496, -13524000, 3622080, -564480, 40320

Offset: 0

Peter Luschny, May 27 2017

The zeta polynomials for the poset P_n of ordered pairs (S,T) where S,T are subsets of [n] with |S| = |T| ordered component-wise by inclusion. - Geoffrey Critzer, Jan 22 2021

			Triangle starts:
[0] 1
[1] 0,      1
[2] 0,     -1,       2
[3] 0,      4,      -9,       6
[4] 0,    -33,      82,     -72,      24
[5] 0,    456,   -1225,    1250,    -600,    120
[6] 0,  -9460,   27041,  -30600,   17700,  -5400,    720
[7] 0, 274800, -826336, 1011017, -661500, 249900, -52920, 5040
...
For example let p4(x) = -33*x + 82*x^2 - 72*x^3 + 24*x^4 then p4(n) = A169712(n).

Cf. A287316, A000384 (p2), A169711 (p3), A169712 (p4), A169713 (p5).

Cf. A000275(n), A212855.

A287314_row := proc(n) local k; sum(z^k/k!^2, k = 0..infinity);
series(%^x, z=0, n+1): n!^2*coeff(%,z,n); seq(coeff(%,x,k), k=0..n) end:
for n from 0 to 8 do print(A287314_row(n)) od;
A287314_poly := proc(n) local k, x; sum(z^k/k!^2, k = 0..infinity);
series(%^x, z=0, n+1): unapply(n!^2*coeff(%, z, n), x) end:
for n from 0 to 7 do A287314_poly(n) od;

nn = 10; e[x_] := Sum[x^n/n!^2, {n, 0, nn}];
f[list_] := CoefficientList[InterpolatingPolynomial[Table[{i, list[[i]]}, {i, 1, nn}], m], m];Drop[Map[f,Transpose[Table[Table[n!^2, {n, 0, nn}] CoefficientList[
Series[e[x]^k, {x, 0, nn}], x], {k, 1, nn}]]], -1] // Grid (* Geoffrey Critzer, Jan 22 2021 *)

Sum_{k=0..n} abs(T(n,k)) = A000275(n) = A212855_row(2).

A287315 Triangle read by rows, (Sum_{k=0..n} T[n,k]*x^k) / (1-x)^(n+1) are generating functions of the columns of A287316.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 16, 19, 0, 1, 65, 299, 211, 0, 1, 246, 3156, 7346, 3651, 0, 1, 917, 28722, 160322, 237517, 90921, 0, 1, 3424, 245407, 2864912, 9302567, 9903776, 3081513, 0, 1, 12861, 2041965, 46261609, 288196659, 632274183, 520507423, 136407699

Offset: 0

Peter Luschny, May 29 2017

			Triangle starts:
0: [1]
1: [0, 1]
2: [0, 1,    3]
3: [0, 1,   16,     19]
4: [0, 1,   65,    299,     211]
5: [0, 1,  246,   3156,    7346,    3651]
6: [0, 1,  917,  28722,  160322,  237517,   90921]
7: [0, 1, 3424, 245407, 2864912, 9302567, 9903776, 3081513]
...
Let q4(x) = (x + 65*x^2 + 299*x^3 + 211*x^4) / (1-x)^5 then the coefficients of the series expansion of q4 give A169712, which is column 4 of A287316.

T(n,n) = A000275(n).

Cf. A192721 (variant), A001044, A287314, A287316.

Delta := proc(a, n) local del, A, u;
A := [seq(a(j), j=0..n+1)]; del := (a, k) -> `if`(k=0, a(0), a(k)-a(k-1));
for u from 0 to n do A := [seq(del(k -> A[k+1], j), j=0..n)] od end:
A287315_row := n -> Delta(A287314_poly(n), n):
for n from 0 to 7 do A287315_row(n) od;
A287315_eulerian := (n,x) -> add(A287315_row(n)[k+1]*x^k,k=0..n)/(1-x)^(n+1):
for n from 0 to 4 do A287315_eulerian(n,x) od;

Sum_{k=0..n} T(n,k) = A001044(n).

A033935 Sum of squares of coefficients in full expansion of (z1+z2+...+zn)^n.

Original entry on oeis.org

1, 1, 6, 93, 2716, 127905, 8848236, 844691407, 106391894904, 17091486402849, 3410496772665940, 827540233598615691, 239946160014513220896, 81932406267721802925925, 32541656017173091541743368, 14874686717916861528415671285, 7753005946480818323895940923376

Offset: 0

Warren D. Smith, Dec 11 1999

Two samples of size n are taken from an urn containing infinitely many marbles of n distinct colors. a(n)/n^(2*n) is the probability that the two samples match. That is, they contain the same number of each color of marbles without regard to order. - Geoffrey Critzer, Apr 19 2014

Alois P. Heinz, Table of n, a(n) for n = 0..237

Column k=2 of A245397.
Main diagonal of A287316.
Cf. A364116.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 21 2014
A033935:= proc(n) series(hypergeom([],[1],z)^n, z=0, n+1): n!^2*coeff(%,z,n) end: seq(A033935(n), n=0..16); # Peter Luschny, May 31 2017

Table[nn=n;n!^2 Coefficient[Series[(Sum[x^k/k!^2,{k,0,nn}])^n,{x,0,nn}],x^n],{n,1,20}] (* Geoffrey Critzer, Apr 19 2014 *)
Flatten[{1,Table[n!^2*Coefficient[Series[BesselI[0,2*Sqrt[x]]^n,{x,0,n}],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Jul 29 2014 *)
Table[SeriesCoefficient[HypergeometricPFQ[{},{1},x]^n, {x,0,n}] n!^2, {n,0,16}] (* Peter Luschny, May 31 2017 *)

a(n) is coefficient of x^n in expansion of n!^2*(1 + x/1!^2 + x^2/2!^2 + x^3/3!^2 + ... + x^n/n!^2)^n. - Vladeta Jovovic, Jun 09 2000
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 2.1024237701057210364324371415246345951600138303179762223318873762632384990..., c = 0.487465475752598098146353111500372156824276600165331887960705498284416... - Vaclav Kotesovec, Jul 29 2014, updated Jul 10 2023
a(n) = n!^2 * [z^n] hypergeom([], [1], z)^n. - Peter Luschny, May 31 2017

More terms from James Sellers, Jun 01 2000 and Vladeta Jovovic, Jun 05 2000

a(0)=1 inserted by Alois P. Heinz, Jul 21 2014

A287318 Square array A(n,k) = (2n)! [x^n] BesselI(0, 2sqrt(x))^k read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 36, 20, 0, 1, 8, 90, 400, 70, 0, 1, 10, 168, 1860, 4900, 252, 0, 1, 12, 270, 5120, 44730, 63504, 924, 0, 1, 14, 396, 10900, 190120, 1172556, 853776, 3432, 0, 1, 16, 546, 19920, 551950, 7939008, 32496156, 11778624, 12870, 0

Offset: 0

Peter Luschny, May 23 2017

			Arrays start:
  k\n| 0   1    2      3        4          5           6
  ---|---------------------------------------------------------
  k=0| 1,  0,   0,     0,       0,         0,            0, ... A000007
  k=1| 1,  2,   6,    20,      70,       252,          924, ... A000984
  k=2| 1,  4,  36,   400,    4900,     63504,       853776, ... A002894
  k=3| 1,  6,  90,  1860,   44730,   1172556,     32496156, ... A002896
  k=4| 1,  8, 168,  5120,  190120,   7939008,    357713664, ... A039699
  k=5| 1, 10, 270, 10900,  551950,  32232060,   2070891900, ... A287317
  k=6| 1, 12, 396, 19920, 1281420,  96807312,   8175770064, ... A356258
  k=7| 1, 14, 546, 32900, 2570050, 238935564,  25142196156, ...
  k=8| 1, 16, 720, 50560, 4649680, 514031616,  64941883776, ...
  k=9| 1, 18, 918, 73620, 7792470, 999283068, 147563170524, ...

Nikolai Beluhov, Powers of 2 in High-Dimensional Lattice Walks, arXiv:2506.12789 [math.CO], 2025. See p. 19.

Rows: A000007 (k=0), A000984 (k=1), A002894 (k=2), A002896 (k=3), A039699 (k=4), A287317 (k=5), A356258 (k=6).
Columns: A005843 (n=1), A152746 (n=2), 20*A169711 (n=3), 70*A169712 (n=4), 252*A169713 (n=5).
Main diagonal gives A303503.
Cf. A287316.

A287318_row := proc(k, len) local b, ser;
b := k -> BesselI(0, 2*sqrt(x))^k: ser := series(b(k), x, len);
seq((2*i)!*coeff(ser,x,i), i=0..len-1) end:
for k from 0 to 6 do A287318_row(k, 9) od;

Table[Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^k, {x, 0, n}] (2 n)!, {n, 0, 6}], {k, 0, 6}]

A(n,k) = A287316(n,k) * binomial(2*n,n).

A287698 Square array A(n,k) = (n!)^3 [x^n] hypergeom([], [1, 1], z)^k read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 10, 1, 0, 1, 4, 27, 56, 1, 0, 1, 5, 52, 381, 346, 1, 0, 1, 6, 85, 1192, 6219, 2252, 1, 0, 1, 7, 126, 2705, 36628, 111753, 15184, 1, 0, 1, 8, 175, 5136, 124405, 1297504, 2151549, 104960, 1, 0

Offset: 0

Peter Luschny, May 30 2017

Let A_m(n,k) = (n!)^m [x^n] hypergeom([], [1,…,1], z)^k where [1,…,1] lists (m-1) times 1. These arrays can be seen as generalizations of the power functions n^k. For m = 1 -> A003992, m = 2 -> A287316, m = 3 -> A287698.
A_m(n,n) is the sum of m-th powers of coefficients in the full expansion of (z_1+z_2+...+z_n)^n (compare A245397).
A287696 provide polynomials and A287697 rational functions generating the columns of the array.

			Array starts:
k\n| 0  1    2       3       4         5           6             7
---|-------------------------------------------------------------------
k=0| 1, 0,   0,      0,      0,        0,          0,            0, ... A000007
k=1| 1, 1,   1,      1,      1,        1,          1,            1, ... A000012
k=2| 1, 2,  10,     56,    346,     2252,      15184,       104960, ... A000172
k=3| 1, 3,  27,    381,   6219,   111753,    2151549,     43497891, ... A141057
k=4| 1, 4,  52,   1192,  36628,  1297504,   50419096,   2099649808, ... A287699
k=5| 1, 5,  85,   2705, 124405,  7120505,  464011825,  33031599725, ...
k=6| 1, 6, 126,   5136, 316206, 25461756, 2443835736, 263581282656, ...
       A001107,A287702,A287700,  A287701,                               A055733

Rows: A000007 (k=0), A000012 (k=1), A000172 (k=2), A141057 (k=3), A287699 (k=4).
Columns: A000172 (n=1), A001477(n=1), A001107 (n=2), A287702 (n=3), A287700 (n=4), A287701 (n=5).
Cf. A055733 (diagonal), A287696, A287697, A003992, A287316, A245397.

A287698_row := (k, len) -> seq(A287696_poly(j)(k), j=0..len):
A287698_row := proc(k, len) hypergeom([], [1, 1], x):
series(%^k, x, len); seq((i!)^3*coeff(%, x, i), i=0..len-1) end:
for k from 0 to 6 do A287698_row(k, 9) od;
A287698_col := proc(n, len) local k, x; hypergeom([], [1, 1], z);
series(%^x, z=0, n+1): unapply(n!^3*coeff(%, z, n), x); seq(%(j), j=0..len) end:
for n from 0 to 7 do A287698_col(n, 9) od;

Table[Table[SeriesCoefficient[HypergeometricPFQ[{},{1,1},x]^k, {x, 0, n}] (n!)^3, {n, 0, 6}], {k, 0, 9}] (* as a table of rows *)

A361397 Number A(n,k) of k-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 20, 4, 0, 1, 8, 54, 176, 10, 0, 1, 10, 104, 996, 1876, 28, 0, 1, 12, 170, 2944, 22734, 22064, 84, 0, 1, 14, 252, 6500, 108136, 577692, 275568, 264, 0, 1, 16, 350, 12144, 332050, 4525888, 15680628, 3584064, 858, 0

Offset: 0

Alois P. Heinz, Mar 10 2023

Column k is INVERTi transform of k-th row of A287318.

			Square array A(n,k) begins:
  1,  1,     1,      1,       1,        1,        1, ...
  0,  2,     4,      6,       8,       10,       12, ...
  0,  2,    20,     54,     104,      170,      252, ...
  0,  4,   176,    996,    2944,     6500,    12144, ...
  0, 10,  1876,  22734,  108136,   332050,   796860, ...
  0, 28, 22064, 577692, 4525888, 19784060, 62039088, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-5 give: A000007, |A002420|, A054474, A049037, A359801, A361364.
Rows n=0-2 give: A000012, A005843, A139271.
Main diagonal gives A361297.
Cf. A000108, A287318.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
    end:
g:= proc(n, k) option remember; `if` (n<1, -1,
      -add(g(n-i, k)*(2*i)!*b(i, k)/i!^2, i=1..n))
    end:
A:= (n,k)-> `if`(n=0, 1, `if`(k=0, 0, g(n, k))):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, 0] = 0; b[n_, 1] = 1; b[0, k_] = 1;
b[n_, k_] := b[n, k] = Sum[Binomial[n, i]^2*b[i, k - 1], {i, 0, n}]; (* A287316 *)
g[n_, k_] := g[n, k] = b[n, k]*Binomial[2 n, n]; (* A287318 *)
a[n_, k_] := a[n, k] = g[n, k] - Sum[a[i, k]*g[n - i, k], {i, 1, n - 1}];
TableForm[Table[a[n, k], {k, 0, 10}, {n, 0, 10}]] (* Shel Kaphan, Mar 14 2023 *)

A(n,1)/2 = A000108(n-1) for n >= 1.

G.f. of column k: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^k dt. - Shel Kaphan, Mar 19 2023

A287317 Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages.

Original entry on oeis.org

1, 10, 270, 10900, 551950, 32232060, 2070891900, 142317232200, 10277494548750, 770878551371500, 59577647564312020, 4717432065143561400, 381091087190569291900, 31308955091335405435000, 2609450031306515140215000, 220199552765301571338488400

Offset: 0

Peter Luschny, May 23 2017

Case k=5 of A287318.
Cf. A169714, A287316
1-4 dimensional analogs are A000984, A002894, A002896, A039699.

A287317_list := proc(len) series(BesselI(0, 2*sqrt(x))^5, x, len);
seq((2*i)!*coeff(%, x, i), i=0..len-1) end: A287317_list(16);

Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^5, {x, 0, n}] (2 n) !, {n, 0, 15}]
Table[Binomial[2n,n]^2 Sum[(Binomial[n,j]^4/Binomial[2n,2j]) HypergeometricPFQ[{-j,-j,-j}, {1,1/2-j}, 1/4], {j,0,n}], {n,0,15}]
Table[Sum[(2 n)!/(i! j! k! l! (n-i-j-k-l)!)^2, {i,0,n}, {j,0,n-i}, {k,0,n-i-j}, {l,0,n-i-j-k}], {n,0,30}] (* Shel Kaphan, Jan 24 2023 *)

a(n) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^5.
a(n) = binomial(2*n,n)*A169714(n).
a(n) ~ 2^(2*n) * 5^(2*n + 5/2) / (16 * Pi^(5/2) * n^(5/2)). - Vaclav Kotesovec, Nov 13 2017
a(n) = Sum_{i+j+k+l+m=n, 0<=i,j,k,l,m<=n} multinomial(2n, [i,i,j,j,k,k,l,l,m,m]). - Shel Kaphan, Jan 24 2023

Moved original definition to formula section and reworded definition descriptively similar to sequence A039699, by Dave R.M. Langers, Oct 12 2022

A169712 The function W_n(8) (see Borwein et al. reference for definition).

Original entry on oeis.org

1, 70, 639, 2716, 7885, 18306, 36715, 66424, 111321, 175870, 265111, 384660, 540709, 740026, 989955, 1298416, 1673905, 2125494, 2662831, 3296140, 4036221, 4894450, 5882779, 7013736, 8300425, 9756526, 11396295, 13234564, 15286741, 17568810, 20097331, 22889440

Offset: 1

N. J. A. Sloane, Apr 17 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jonathan M. Borwein, Dirk Nuyens, Armin Straub, and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, May 2011.
Pakawut Jiradilok and Elchanan Mossel, Gaussian Broadcast on Grids, arXiv:2402.11990 [cs.IT], 2024. See p. 27.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Column 4 of A287316.

Cf. A287314.

[-33*n+82*n^2-72*n^3+24*n^4: n in [1..40]]; // Vincenzo Librandi May 28 2017

A169712 := proc(n)
        W(n,8) ;
end proc:
seq(A169712(n),n=1..40) ; # uses W defined in A169715; R. J. Mathar, Mar 28 2012
a := n -> -33*n + 82*n^2 - 72*n^3 + 24*n^4:
seq(a(n), n=1..28); # Peter Luschny, May 27 2017

Table[-33 n + 82 n^2 - 72 n^3 + 24 n^4, {n, 1, 40}] (* or *) CoefficientList[Series[(1 + 65 x + 299 x^2 + 211 x^3) /(1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2017 *)

a(n)=-33*n+82*n^2-72*n^3+24*n^4 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = -33*n + 82*n^2 - 72*n^3 + 24*n^4. - Peter Luschny, May 27 2017
G.f.: x*(1+65*x+299*x^2+211*x^3)/(1-x)^5. - Vincenzo Librandi, May 28 2017
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Vincenzo Librandi, May 28 2017

A169711 The function W_n(6) (see Borwein et al. reference for definition).

Original entry on oeis.org

1, 20, 93, 256, 545, 996, 1645, 2528, 3681, 5140, 6941, 9120, 11713, 14756, 18285, 22336, 26945, 32148, 37981, 44480, 51681, 59620, 68333, 77856, 88225, 99476, 111645, 124768, 138881, 154020, 170221, 187520, 205953, 225556, 246365, 268416, 291745, 316388

Offset: 1

N. J. A. Sloane, Apr 17 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jonathan M. Borwein, Dirk Nuyens, Armin Straub, and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, May 2011.
Pakawut Jiradilok and Elchanan Mossel, Gaussian Broadcast on Grids, arXiv:2402.11990 [cs.IT], 2024. See p. 27.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

The sequence in Table 1 of the Borwein et al. reference are A000384, A109711-A109713; A000984, A002893, A002895, A169714, A169715.
Column 3 of A287316.
Cf. A287314.

[6*n^3-9*n^2+4*n: n in [1..40]]; // Vincenzo Librandi, May 28 2017

A169711 := proc(n)
        W(n,6) ;
end proc:
seq(A169711(n),n=1..20) ; # uses W from A169715; R. J. Mathar, Mar 28 2012
a := n -> 6*n^3 - 9*n^2 + 4*n: seq(a(n), n=1..33); # Peter Luschny, May 27 2017

CoefficientList[Series[(1 + 16 x + 19 x^2) / (1 - x)^4, {x, 0, 50}], x] (* or *) Table[6 n^3 - 9 n^2 + 4 n, {n, 1, 40}] (* Vincenzo Librandi, May 28 2017 *)
LinearRecurrence[{4,-6,4,-1},{1,20,93,256},40] (* Harvey P. Dale, Feb 27 2023 *)

a(n)=6*n^3-9*n^2+4*n \\ Charles R Greathouse IV, Oct 18 2022

a(n) = 6*n^3 - 9*n^2 + 4*n. - Peter Luschny, May 27 2017
G.f.: x*(1+16*x+19*x^2)/(1-x)^4. - Vincenzo Librandi, May 28 2017
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, May 28 2017

A169713 The function W_n(10) (see Borwein et al. reference for definition).

Original entry on oeis.org

1, 252, 4653, 31504, 127905, 384156, 948157, 2039808, 3965409, 7132060, 12062061, 19407312, 29963713, 44685564, 64699965, 91321216, 126065217, 170663868, 227079469, 297519120, 384449121, 490609372, 619027773, 773034624, 956277025, 1172733276

Offset: 1

N. J. A. Sloane, Apr 17 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, May 2011.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Column 5 of A287316.

Cf. A287314.

[120*n^5-600*n^4+1250*n^3-1225*n^2+456*n: n in [1..40]]; // Vincenzo Librandi, May 28 2017

A169713 := proc(n)
        W(n,10) ;
end proc:
seq(A169713(n),n=1..20) ; # uses W() from A169715; R. J. Mathar, Mar 27 2012
a := n -> 120*n^5 - 600*n^4 + 1250*n^3 - 1225*n^2 + 456*n:
seq(a(n), n=1..20); # Peter Luschny, May 27 2017

Table[120 n^5 - 600 n^4 + 1250 n^3 - 1225 n^2 + 456 n, {n, 1, 40}] (* or *) CoefficientList[Series[(1 + 246 x + 3156 x^2 + 7346 x^3 + 3651 x^4) / (1 - x)^6, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2017 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{1,252,4653,31504,127905,384156},30] (* Harvey P. Dale, Aug 09 2023 *)

a(n)=120*n^5-600*n^4+1250*n^3-1225*n^2+456*n \\ Charles R Greathouse IV, Oct 21 2022

a(n) = 120*n^5 - 600*n^4 + 1250*n^3 - 1225*n^2 + 456*n. - Peter Luschny, May 27 2017
G.f.: x*(1+246*x+3156*x^2+7346*x^3+3651*x^4)/(1-x)^6. - Vincenzo Librandi, May 28 2017
a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Vincenzo Librandi, May 28 2017

cp's OEIS Frontend

A287314 Triangle read by rows, the coefficients of the polynomials generating the columns of A287316.

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A287315 Triangle read by rows, (Sum_{k=0..n} T[n,k]*x^k) / (1-x)^(n+1) are generating functions of the columns of A287316.

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A033935 Sum of squares of coefficients in full expansion of (z1+z2+...+zn)^n.

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A287318 Square array A(n,k) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.

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A287698 Square array A(n,k) = (n!)^3 [x^n] hypergeom([], [1, 1], z)^k read by antidiagonals.

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A361397 Number A(n,k) of k-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A287317 Number of 5-dimensional cubic lattice walks that start and end at origin after 2n steps, free to pass through origin at intermediate stages.

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A169712 The function W_n(8) (see Borwein et al. reference for definition).

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A287318 Square array A(n,k) = (2n)! [x^n] BesselI(0, 2sqrt(x))^k read by antidiagonals.