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 1-3 of 3 results.

A287316 Square array A(n,k) = (n!)^2 [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 20, 1, 0, 1, 5, 28, 93, 70, 1, 0, 1, 6, 45, 256, 639, 252, 1, 0, 1, 7, 66, 545, 2716, 4653, 924, 1, 0, 1, 8, 91, 996, 7885, 31504, 35169, 3432, 1, 0, 1, 9, 120, 1645, 18306, 127905, 387136, 272835, 12870, 1, 0
Offset: 0

Views

Author

Peter Luschny, May 23 2017

Keywords

Comments

A287314 provide polynomials and A287315 rational functions generating the columns of the array.

Examples

			Arrays start:
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,   6,     20,      70,     252,       924,       3432, ... A000984
k=3| 1, 3,  15,     93,     639,    4653,     35169,     272835, ... A002893
k=4| 1, 4,  28,    256,    2716,   31504,    387136,    4951552, ... A002895
k=5| 1, 5,  45,    545,    7885,  127905,   2241225,   41467725, ... A169714
k=6| 1, 6,  66,    996,   18306,  384156,   8848236,  218040696, ... A169715
k=7| 1, 7,  91,   1645,   36715,  948157,  27210169,  844691407, ...
k=8| 1, 8, 120,   2528,   66424, 2039808,  70283424, 2643158400, ... A385286
k=9| 1, 9, 153,   3681,  111321, 3965409, 159700401, 7071121017, ...
       A000384,A169711, A169712, A169713,                            A033935

Links

Crossrefs

Rows: A000007 (k=0), A000012 (k=1), A000984 (k=2), A002893 (k=3), A002895 (k=4), A169714 (k=5), A169715 (k=6), A385286 (k=8).
Columns: A001477(n=1), A000384 (n=2), A169711 (n=3), A169712 (n=4), A169713 (n=5).
Cf. A033935 (diagonal), A287314, A287315, A287318.

Programs

  • Maple
    A287316_row := proc(k, len) local b, ser;
b := k -> BesselI(0, 2*sqrt(x))^k: ser := series(b(k), x, len);
seq((i!)^2*coeff(ser,x,i), i=0..len-1) end:
for k from 0 to 6 do A287316_row(k, 9) od;
A287316_col := proc(n, len) local k, x;
sum(z^k/k!^2, k = 0..infinity); series(%^x, z=0, n+1):
unapply(n!^2*coeff(%, z, n), x); seq(%(j), j=0..len) end:
for n from 0 to 7 do A287316_col(n, 9) od;
  • Mathematica
    Table[Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^k, {x, 0, n}] (n!)^2, {n, 0, 6}], {k, 0,9}]
  • PARI
    A287316_row(K, N) = {
  my(x='x + O('x^(2*N-1)));
  Vec(serlaplace(serlaplace(substpol(besseli(0,2*x)^K, 'x^2, 'x))));
};
N=8; concat([vector(N, n, n==1)], vector(9, k, A287316_row(k, N))) \\ Gheorghe Coserea, Jan 12 2018
  • PARI
    {A(n, k) = if(n<0 || k<0, 0, n!^2 * polcoeff(besseli(0, 2*x + x*O(x^(2*n)))^k, 2*n))}; /* Michael Somos, Dec 30 2021 */
  • PARI
    A(k, n) = my(x='x+O('x^(n+1))); n!^2*polcoeff(hypergeom([], [1], x)^k, n); \\ Peter Luschny, Jun 24 2025
  • Python
    from math import comb
from functools import lru_cache
@lru_cache(maxsize=None)
def A(n,k):
    if k <= 0: return 0**n
    return sum(A(i,k-1)*comb(n,i)**2 for i in range(n+1))
for k in range(10): print([A(n, k) for n in range(8)])
# Jeremy Tan, Dec 10 2021

Formula

A(n,k) = A287318(n,k) / binomial(2*n,n).
If a+b=k then A(n,k) = Sum_{i=0..n} A(i,a)*A(n-i,b)*binomial(n,i)^2 (Richmond and Shallit). In particular A(n,k) = Sum_{i=0..n} A(i,k-1)*binomial(n,i)^2. - Jeremy Tan, Dec 10 2021

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

Views

Author

Peter Luschny, May 23 2017

Keywords

Examples

			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, ...

Links

Crossrefs

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.

Programs

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

Formula

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

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

Original entry on oeis.org

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

Views

Author

Peter Luschny, May 27 2017

Keywords

Comments

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

Examples

			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).

Crossrefs

Cf. A287316, A000384 (p2), A169711 (p3), A169712 (p4), A169713 (p5).
Cf. A000275(n), A212855.

Programs

  • Maple
    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;
  • Mathematica
    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 *)

Formula

Sum_{k=0..n} abs(T(n,k)) = A000275(n) = A212855_row(2).
Showing 1-3 of 3 results.

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