A271453 Triangle read by rows of coefficients of polynomials C_n(x) = Sum_{k=0..n} (2*k)!*(x - 1)^(n-k)/((k + 1)!*k!).

1, 0, 1, 2, -1, 1, 3, 3, -2, 1, 11, 0, 5, -3, 1, 31, 11, -5, 8, -4, 1, 101, 20, 16, -13, 12, -5, 1, 328, 81, 4, 29, -25, 17, -6, 1, 1102, 247, 77, -25, 54, -42, 23, -7, 1, 3760, 855, 170, 102, -79, 96, -65, 30, -8, 1, 13036, 2905, 685, 68, 181, -175, 161, -95, 38, -9, 1, 45750, 10131, 2220, 617, -113, 356, -336, 256, -133, 47, -10, 1

Offset: 0

Author

Ilya Gutkovskiy, Apr 09 2016

Keywords

• sign
• tabl
• easy

Comments

The polynomials C_n(x) have generating function G(x,t) = (1 - sqrt(1 - 4*t))/(2*t*(1 + t - x*t)) = 1 + x*t + (x^2 - x + 2)*t^2 + (x^3 - 2*x^2 + 3*x + 3)*t^3 + ...

C_n(x) can be defined by the recurrence relation C_n(x) = (x - 1)*C_(n-1)(x) + (2n)!/((n + 1)!*n!), C_0(x) = 1 or the equivalent form C_n(x) = (x - 1)*C_(n-1)(x) + C_n(1), C_0(x) = 1.

C_n(x) can be defined as convolution of Catalan numbers and powers of (x - 1).

Discriminants of C_n(x) gives the sequence: 1, 1, -7, -543, 533489, 7080307052, -1318026434480736, -3526797951451513832247, 137992774365121594001729513153, ...

C_n(0) = A032357(n).

C_n(1) = C_n(x) - (x - 1)*C_(n-1)(x) = A000108(n).

C_n(2) = Sum_{m=0..n} C_1(m) = A014137(n).

C_n(3) = A014318(n).

C_n(5) = A000346(n).

C_n(6) = A046714(n).

Examples

			Triangle begins:
   1;
   0,  1;
   2, -1,  1;
   3,  3, -2,  1;
  11,  0,  5, -3,  1;
  31, 11, -5,  8, -4,  1;
  ...
The first few polynomials are:
  C_0(x) = 1;
  C_1(x) = x;
  C_2(x) = x^2 -   x   + 2;
  C_3(x) = x^3 - 2*x^2 + 3*x   + 3;
  C_4(x) = x^4 - 3*x^3 + 5*x^2         + 11;
  C_5(x) = x^5 - 4*x^4 + 8*x^3 - 5*x^2 + 11*x + 31;
  ...
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened
• Ilya Gutkovskiy, Polynomials C_n(x)
• Eric Weisstein's World of Mathematics, Catalan Number

Crossrefs

Cf. A000108, A130595.

Programs

Mathematica

CoefficientList[RecurrenceTable[{c[0] == 1, c[n] == (x - 1) c[n - 1] + CatalanNumber[n]}, c, {n, 11}], x]
T[n_, n_]:= 1; T[n_, 0]:= (-1)^n*Sum[CatalanNumber[k]*(-1)^k, {k, 0, n}]; T[n_, k_]:= T[n - 1, k - 1] - T[n - 1, k]; Table[T[n, k], {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 04 2018 *)

PARI

{T(n, k) = if(k==n, 1, if(k==0, sum(j=0,n, (-1)^(n-j)*(2*j)!/(j!*(j+1)!)), T(n-1, k-1) - T(n-1, k))) };
for(n=0, 10, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 04 2018

Formula

For triangle: T(n,n)=1, T(n,0) = Sum_{k=0..n} (-1)^(n-k)*(2*k)!/(k! * (k+1)!), T(n, k) = T(n-1, k-1) - T(n-1, k). - G. C. Greubel, Nov 04 2018