A049410 A triangle of numbers related to triangle A049325.
1, 3, 1, 6, 9, 1, 6, 51, 18, 1, 0, 210, 195, 30, 1, 0, 630, 1575, 525, 45, 1, 0, 1260, 10080, 6825, 1155, 63, 1, 0, 1260, 51660, 71505, 21840, 2226, 84, 1, 0, 0, 207900, 623700, 333585, 57456, 3906, 108, 1, 0, 0, 623700, 4573800, 4293135, 1195425, 131670
Offset: 1
Examples
Triangle begins:
{1};
{3,1};
{6,9,1};
{6,51,18,1};
...
E.g. row polynomial E(3,x)= 6*x+9*x^2+x^3.
Links
- Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Crossrefs
Row sums give A049426.
Programs
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Mathematica
rows = 10; t = Table[Product[4k+3, {k, 0, n-1}], {n, 0, rows}]; T[n_, k_] := BellY[n, k, t]; M = Inverse[Array[T, {rows, rows}]] // Abs; A049325 = Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *) -
Sage
# uses[inverse_bell_transform from A265605] # Adds a column 1,0,0,0,... at the left side of the triangle. multifact_4_3 = lambda n: prod(4*k + 3 for k in (0..n-1)) inverse_bell_matrix(multifact_4_3, 9) # Peter Luschny, Dec 31 2015
Formula
a(n, m) = n!*A049325(n, m)/(m!*4^(n-m)); a(n, m) = (4*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n
A049349 Row sums of triangle A049325.
1, 7, 29, 103, 405, 1599, 6141, 23863, 92773, 359791, 1396493, 5421415, 21041397, 81670431, 317005341, 1230432919, 4775854213, 18537264079, 71951401517, 279275580103, 1083993881877, 4207466012031, 16331061009213
Offset: 1
Comments
p(3,x) is row polynomial corresponding to triangle row A033842(3,m).
Links
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Formula
G.f.: x*(1+6*x+16*x^2+16*x^3)/(1-x-6*x^2-16*x^3-16*x^4) = x*p(3, x)/(1-x*p(3, x)) with x*p(3, x) G.f. for first column of A049325.
A049323 Triangle of coefficients of certain polynomials (exponents in increasing order), equivalent to A033842.
1, 1, 1, 1, 3, 3, 1, 6, 16, 16, 1, 10, 50, 125, 125, 1, 15, 120, 540, 1296, 1296, 1, 21, 245, 1715, 7203, 16807, 16807, 1, 28, 448, 4480, 28672, 114688, 262144, 262144, 1, 36, 756, 10206, 91854, 551124, 2125764, 4782969, 4782969, 1, 45, 1200, 21000, 252000
Offset: 0
Comments
These polynomials p(n, x) appear in the W. Lang reference as c1(-(n+1);x), n >= 0 on p.12. The coefficients are given there in eq.(44) on p. 6. - Wolfdieter Lang, Nov 20 2015
Examples
The triangle a(n, m) begins:
n\m 0 1 2 3 4 5 6 7 ...
0: 1
1: 1 1
2: 1 3 3
3: 1 6 16 16
4: 1 10 50 125 125
5: 1 15 120 540 1296 1296
6: 1 21 245 1715 7203 16807 16807
7: 1 28 448 4480 28672 114688 262144 262144
... reformatted. - Wolfdieter Lang, Nov 20 2015
E.g. the third row {1,3,3} corresponds to polynomial p(2,x)= 1 + 3*x + 3*x^2.
Links
- W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Crossrefs
a(n, 0)= A000012 (powers of 1), a(n, 1)= A000217 (triangular numbers), a(n, n)= A000272(n+1), n >= 0 (diagonal), a(n, n-1)= A000272(n+1), n >= 1.
Programs
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Magma
/* As triangle: */ [[Binomial(n+1, k+1)*(n+1)^(k-1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 20 2015
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Maple
seq(seq(binomial(n+1,m+1)*(n+1)^(m-1),m=0..n),n=0..10); # Robert Israel, Oct 19 2015
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Mathematica
Table[Binomial[n + 1, k + 1] (n + 1)^(k - 1), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 19 2015 *)
Formula
a(n, m) = A033842(n, n-m) = binomial(n+1, m+1)*(n+1)^{m-1}, n >= m >= 0, else 0.
p(k-1, -x)/(1-k*x)^k =(-1+1/(1-k*x)^k)/(x*k^2) is for k=1..5 G.f. for A000012, A001792, A036068, A036070, A036083, respectively.
From Werner Schulte, Oct 19 2015: (Start)
a(2*n,n) = A000108(n)*(2*n+1)^n;
a(3*n,2*n) = A001764(n)*(3*n+1)^(2*n);
a(p*n,(p-1)*n) = binomial(p*n,n)/((p-1)*n+1)*(p*n+1)^((p-1)*n) for p > 0;
Sum_{m=0..n} (m+1)*a(n,m) = (n+2)^n;
Sum_{m=0..n} (-1)^m*(m+1)*a(n,m) = (-n)^n where 0^0 = 1;
p(n,x) = Sum_{m=0..n} a(n,m)*x^m = ((1+(n+1)*x)^(n+1)-1)/((n+1)^2*x).
(End)
Comments