The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A075499 Stirling2 triangle with scaled diagonals (powers of 4). 14
 1, 4, 1, 16, 12, 1, 64, 112, 24, 1, 256, 960, 400, 40, 1, 1024, 7936, 5760, 1040, 60, 1, 4096, 64512, 77056, 22400, 2240, 84, 1, 16384, 520192, 989184, 435456, 67200, 4256, 112, 1, 65536, 4177920, 12390400, 7956480, 1779456, 169344, 7392, 144, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays. The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(4*z) - 1)*x/4) - 1 Also the inverse Bell transform of the quadruple factorial numbers 4^n*n! (A047053) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 FORMULA a(n, m) = (4^(n-m)) * stirling2(n, m). a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*4)^(n-m))/(m-1)! for n >= m >= 1, else 0. a(n, m) = 4m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1. G.f. for m-th column: (x^m)/Product_{k=1..m}(1-4k*x), m >= 1. E.g.f. for m-th column: (((exp(4x)-1)/4)^m)/m!, m >= 1. EXAMPLE [1]; [4,1]; [16,12,1]; ...; p(3,x) = x(16 + 12*x + x^2). From Andrew Howroyd, Mar 25 2017: (Start) Triangle starts *     1 *     4      1 *    16     12      1 *    64    112     24      1 *   256    960    400     40     1 *  1024   7936   5760   1040    60    1 *  4096  64512  77056  22400  2240   84   1 * 16384 520192 989184 435456 67200 4256 112 1 (End) MATHEMATICA Table[(4^(n - m)) StirlingS2[n, m], {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *) PROG (Sage) # uses[inverse_bell_transform from A265605] # Adds a column 1, 0, 0, ... at the left side of the triangle. multifact_4_4 = lambda n: prod(4*k + 4 for k in (0..n-1)) inverse_bell_matrix(multifact_4_4, 9) # Peter Luschny, Dec 31 2015 (PARI) for(n=1, 11, for(m=1, n, print1(4^(n - m) * stirling(n, m, 2), ", "); ); print(); ) \\ Indranil Ghosh, Mar 25 2017 CROSSREFS Columns 1-7 are A000302, A016152, A019677, A075907-A075910. Row sums are A004213. Cf. A075498, A075500. Sequence in context: A271262 A292922 A117438 * A099394 A269698 A059991 Adjacent sequences:  A075496 A075497 A075498 * A075500 A075501 A075502 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Oct 02 2002 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 28 03:43 EST 2021. Contains 340490 sequences. (Running on oeis4.)