A005573 -id:A005573 - cp's OEIS frontend

A292630 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(kx)(BesselI(0,2x) + BesselI(1,2x)).

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 10, 13, 6, 1, 5, 17, 35, 35, 10, 1, 6, 26, 75, 126, 96, 20, 1, 7, 37, 139, 339, 462, 267, 35, 1, 8, 50, 233, 758, 1558, 1716, 750, 70, 1, 9, 65, 363, 1491, 4194, 7247, 6435, 2123, 126, 1, 10, 82, 535, 2670, 9660, 23460, 34016, 24310, 6046, 252, 1, 11, 101, 755, 4451, 19846, 63195, 132339, 160795, 92378, 17303, 462

Offset: 0

Ilya Gutkovskiy, Sep 20 2017

A(n,k) is the k-th binomial transform of A001405 evaluated at n.

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 2)*x^2/2! +  (k^3 + 3*k^2 + 6*k + 3)*x^3/3! + (k^4 + 4*k^3 + 12*k^2 + 12*k + 6)*x^4/4! + ...
Square array begins:
   1,   1,    1,     1,     1,     1,  ...
   1,   2,    3,     4,     5,     6,  ...
   2,   5,   10,    17,    26,    37,  ...
   3,  13,   35,    75,   139,   233,  ...
   6,  35,  126,   339,   758,  1491,  ...
  10,  96,  462,  1558,  4194,  9660,  ...

N. J. A. Sloane, Transforms

Columns k=0..5 give A001405, A005773 (with first term deleted), A001700, A026378 (with offset 0), A005573, A122898.

Main diagonal gives A292631.

[seq(seq((k)!*add((m-j)^(j-i)/floor(i/2)!/ceil(i/2)!/(j-i)!,i=0..j),j=0..m), m=0..20)]; # Robert Israel, Sep 20 2017

Table[Function[k, n! SeriesCoefficient[Exp[k x] (BesselI[0, 2 x] + BesselI[1, 2 x]), {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

E.g.f. of column k: exp(k*x)*(BesselI(0,2*x) + BesselI(1,2*x)).

A349541 E.g.f.: exp(x) * (BesselI(0,8x) + BesselI(1,8x)).

Original entry on oeis.org

1, 5, 41, 301, 2513, 20181, 170745, 1423101, 12161441, 103344037, 889924553, 7650373325, 66271512433, 574065261173, 4996181205657, 43511277885597, 380108373809985, 3323551100483397, 29122753514303337, 255427680480306285, 2243831648555990289, 19728657265135701525

Offset: 0

Ilya Gutkovskiy, Nov 21 2021

nonn

Cf. A001405, A005573, A005773, A151318, A349540.

nmax = 21; CoefficientList[Series[Exp[x] (BesselI[0, 8 x] + BesselI[1, 8 x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] Binomial[k, Floor[k/2]] 4^k, {k, 0, n}], {n, 0, 21}]

a(n) = sum(k=0, n, binomial(n,k) * binomial(k, k\2) * 4^k); \\ Michel Marcus, Nov 21 2021

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(k,floor(k/2)) * 4^k.

a(n) ~ 3^(2*n + 1) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 26 2021

A104855 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of three-dimensional lattice walks consisting of n unit steps, each in one of the six coordinate directions, starting at the origin, never going below the horizontal plane and having k vertical steps.

Original entry on oeis.org

1, 4, 1, 16, 8, 2, 64, 48, 24, 3, 256, 256, 192, 48, 6, 1024, 1280, 1280, 480, 120, 10, 4096, 6144, 7680, 3840, 1440, 240, 20, 16384, 28672, 43008, 26880, 13440, 3360, 560, 35, 65536, 131072, 229376, 172032, 107520, 35840, 8960, 1120, 70, 262144, 589824

Offset: 0

Emeric Deutsch, Apr 23 2005

			T(2,1)=8 because we have NU, SU, EU, WU, UN, US, UE and UW, where N=(0,1,0),S=(0,-1,0), E=(1,0,0),W=(-1,0,0), U=(0,0,1) and S=(0,0,-1).
Triangle begins:
   1;
   4,  1;
  16,  8,  2;
  64, 48, 24,  3;

J. Brawner, Three-Dimensional Lattice Walks in the Upper Half-Space: Problem 10795, Amer. Math. Monthly, 108 (Dec. 2001), 980.

Row sums yield A005573. T(n,n) = A001405(n), T(n,0) = A000302(n) (powers of 4).

Cf. A005573, A001405, A000302.

T:=(n,k)->binomial(n,k)*binomial(k,ceil(k/2))*4^(n-k): for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

T(n, k) = binomial(n, k)*binomial(k, ceiling(k/2))*4^(n-k).

A171814 Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 35, 30, 9, 1, 126, 140, 60, 12, 1, 462, 630, 350, 100, 15, 1, 1716, 2772, 1890, 700, 150, 18, 1, 6435, 12012, 9702, 4410, 1225, 210, 21, 1, 24310, 51480, 48048, 25872, 8820, 1960, 280, 24, 1

Offset: 0

Philippe Deléham, Dec 19 2009

			Triangle begins:
     1;
     3,    1;
    10,    6,    1;
    35,   30,    9,   1;
   126,  140,   60,  12,   1;
   462,  630,  350, 100,  15,  1;
  1716, 2772, 1890, 700, 150, 18, 1;
  ...

Cf. A107230, A171651

Cf. A001405, A001700, A005573, A005773, A026378, A099323, A122898, A168491.

T[n_,k_]:=n!SeriesCoefficient[Exp[2*x]*(BesselI[0,2*x]+BesselI[1,2*x])*x^k / k!,{x,0,n}]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* Stefano Spezia, Dec 23 2023 *)

Sum_{k, 0<=k<=n} T(n,k)*x^k = A168491(n), A099323(n+1), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively.
Conjectural g.f.: 1/(2*t)*( sqrt( (1 - x*t)/(1 - (4 + x)*t) ) - 1 ) = 1 + (3 + x)*t + (10 + 6*x + x^2)*t^2 + .... - Peter Bala, Nov 10 2013
E.g.f. of column k: exp(2*x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - Mélika Tebni, Dec 23 2023

cp's OEIS Frontend

A292630 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)*(BesselI(0,2*x) + BesselI(1,2*x)).

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A349541 E.g.f.: exp(x) * (BesselI(0,8*x) + BesselI(1,8*x)).

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A104855 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of three-dimensional lattice walks consisting of n unit steps, each in one of the six coordinate directions, starting at the origin, never going below the horizontal plane and having k vertical steps.

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Maple

Formula

A171814 Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).

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A292630 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(kx)(BesselI(0,2x) + BesselI(1,2x)).

A349541 E.g.f.: exp(x) * (BesselI(0,8x) + BesselI(1,8x)).