A303487
a(n) = n! * [x^n] 1/(1 - 4*x)^(n/4).
Original entry on oeis.org
1, 1, 12, 231, 6144, 208845, 8648640, 422463195, 23781703680, 1515973484025, 107941254220800, 8491022274509775, 731304510986649600, 68444451854354701125, 6916953288171902976000, 750681472158682148959875, 87076954662428278259712000, 10751175443940144673035200625
Offset: 0
a(1) = 1;
a(2) = 2*6 = 12;
a(3) = 3*7*11 = 231;
a(4) = 4*8*12*16 = 6144;
a(5) = 5*9*13*17*21 = 208845, etc.
Cf.
A000407,
A001813,
A007696,
A008545,
A034176,
A034177,
A047053,
A051617,
A051618,
A051619,
A051620,
A051621,
A051622,
A113551,
A303486,
A303488.
-
Table[n! SeriesCoefficient[1/(1 - 4 x)^(n/4), {x, 0, n}], {n, 0, 17}]
Table[Product[4 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[4^n Pochhammer[n/4, n], {n, 0, 17}]
A370915
A(n, k) = 4^n*Pochhammer(k/4, n). Square array read by ascending antidiagonals.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 5, 2, 1, 0, 45, 12, 3, 1, 0, 585, 120, 21, 4, 1, 0, 9945, 1680, 231, 32, 5, 1, 0, 208845, 30240, 3465, 384, 45, 6, 1, 0, 5221125, 665280, 65835, 6144, 585, 60, 7, 1, 0, 151412625, 17297280, 1514205, 122880, 9945, 840, 77, 8, 1
Offset: 0
The array starts:
[0] 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
[1] 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
[2] 0, 5, 12, 21, 32, 45, 60, 77, 96, ...
[3] 0, 45, 120, 231, 384, 585, 840, 1155, 1536, ...
[4] 0, 585, 1680, 3465, 6144, 9945, 15120, 21945, 30720, ...
[5] 0, 9945, 30240, 65835, 122880, 208845, 332640, 504735, 737280, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
[0] 1;
[1] 0, 1;
[2] 0, 1, 1;
[3] 0, 5, 2, 1;
[4] 0, 45, 12, 3, 1;
[5] 0, 585, 120, 21, 4, 1;
[6] 0, 9945, 1680, 231, 32, 5, 1;
[7] 0, 208845, 30240, 3465, 384, 45, 6, 1;
Columns:
A000007,
A007696,
A001813,
A008545,
A047053,
A007696,
A000407,
A034176,
A052570 and
A034177,
A051617,
A051618,
A051619,
A051620.
-
A := (n, k) -> 4^n*pochhammer(k/4, n):
for n from 0 to 5 do seq(A(n, k), k = 0..9) od;
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 4*x)^(-k/4);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint(EGFcol(n, 9)), n = 0..5);
# Using the generating polynomials for the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-4)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..8)), n = 0..5);
# Implementing the LU decomposition of A:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371026(n-1, k-1)):
U := Matrix(7, 7, (n, k) -> binomial(n-1, k-1)):
MatrixMatrixMultiply(L, Transpose(U));
-
A[n_, k_] := 4^n * Pochhammer[k/4, n]; Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 06 2024 *)
-
def A(n, k): return 4**n * rising_factorial(k/4, n)
for n in range(6): print([A(n, k) for k in range(9)])
A172455
The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.
Original entry on oeis.org
1, 7, 84, 1463, 33936, 990542, 34938624, 1445713003, 68639375616, 3676366634402, 219208706540544, 14397191399702118, 1032543050697424896, 80280469685284582812, 6725557192852592984064, 603931579625379293509683
Offset: 1
G.f. = x + 7*x^2 + 84*x^3 + 1463*x^4 + 33936*x^5 + 990542*x^6 + 34938624*x^7 + ...
a(2) = 7 since (6*2 - 4) * a(2-1) - (a(1) * a(2-1)) = 7.
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
- R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 307.
- R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
- NIST Digital Library of Mathematical Functions, Airy Functions.
- A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
- Eric Weisstein's World of Mathematics, Airy Functions, contains the definitions of Ai(x), Bi(x).
Cf.
A000079 S(1,1,-1),
A000108 S(0,0,1),
A000142 S(1,-1,0),
A000244 S(2,1,-2),
A000351 S(4,1,-4),
A000400 S(5,1,-5),
A000420 S(6,1,-6),
A000698 S(2,-3,1),
A001710 S(1,1,0),
A001715 S(1,2,0),
A001720 S(1,3,0),
A001725 S(1,4,0),
A001730 S(1,5,0),
A003319 S(1,-2,1),
A005411 S(2,-4,1),
A005412 S(2,-2,1),
A006012 S(-1,2,2),
A006318 S(0,1,1),
A047891 S(0,2,1),
A049388 S(1,6,0),
A051604 S(3,1,0),
A051605 S(3,2,0),
A051606 S(3,3,0),
A051607 S(3,4,0),
A051608 S(3,5,0),
A051609 S(3,6,0),
A051617 S(4,1,0),
A051618 S(4,2,0),
A051619 S(4,3,0),
A051620 S(4,4,0),
A051621 S(4,5,0),
A051622 S(4,6,0),
A051687 S(5,1,0),
A051688 S(5,2,0),
A051689 S(5,3,0),
A051690 S(5,4,0),
A051691 S(5,5,0),
A053100 S(6,1,0),
A053101 S(6,2,0),
A053102 S(6,3,0),
A053103 S(6,4,0),
A053104 S(7,1,0),
A053105 S(7,2,0),
A053106 S(7,3,0),
A062980 S(6,-8,1),
A082298 S(0,3,1),
A082301 S(0,4,1),
A082302 S(0,5,1),
A082305 S(0,6,1),
A082366 S(0,7,1),
A082367 S(0,8,1),
A105523 S(0,-2,1),
A107716 S(3,-4,1),
A111529 S(1,-3,2),
A111530 S(1,-4,3),
A111531 S(1,-5,4),
A111532 S(1,-6,5),
A111533 S(1,-7,6),
A111546 S(1,0,1),
A111556 S(1,1,1),
A143749 S(0,10,1),
A146559 S(1,1,-2),
A167872 S(2,-3,2),
A172450 S(2,0,-1),
A172485 S(-1,-2,3),
A177354 S(1,2,1),
A292186 S(4,-6,1),
A292187 S(3, -5, 1).
-
a[1] = 1; a[n_]:= a[n] = (6*n-4)*a[n-1] - Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jan 19 2015 *)
-
{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6 * k - 4) * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */
-
S(v1, v2, v3, N=16) = {
my(a = vector(N)); a[1] = 1;
for (n = 2, N, a[n] = (v1*n+v2)*a[n-1] + v3*sum(j=1,n-1,a[j]*a[n-j])); a;
};
S(6,-4,-1)
\\ test: y = x*Ser(S(6,-4,-1,201)); 6*x^2*y' == y^2 - (2*x-1)*y - x
\\ Gheorghe Coserea, May 12 2017
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