A099927 Pellonomial triangle P(k,n) read by rows.

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 12, 30, 12, 1, 1, 29, 174, 174, 29, 1, 1, 70, 1015, 2436, 1015, 70, 1, 1, 169, 5915, 34307, 34307, 5915, 169, 1, 1, 408, 34476, 482664, 1166438, 482664, 34476, 408, 1, 1, 985, 200940, 6791772, 39618670, 39618670, 6791772, 200940, 985, 1

Offset: 0

Author

Ralf Stephan, Nov 03 2004

Keywords

• nonn
• tabl

Comments

Also (signed) coefficients of solutions to 0 = Sum[i=0..k+1, x(i)*Pell(m+i)^k ].

Sagan and Savage give two combinatorial interpretations for entry T(n,k) in terms of statistics on integer partitions fitting inside a k x (n-k) rectangle. They also relate the values T(n,k) to q-binomial coefficients evaluated at q = -(3 + 2*sqrt(2)). - Peter Bala, Mar 15 2013

Examples

			Triangle starts:
  1;
  1,   1;
  1,   2,    1;
  1,   5,    5,     1;
  1,  12,   30,    12,     1;
  1,  29,  174,   174,    29,    1;
  1,  70, 1015,  2436,  1015,   70,   1;
  1, 169, 5915, 34307, 34307, 5915, 169, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..56, flattened
• Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
• S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
• B. Sagan and C. Savage, Combinatorial Interpretations of Binomial Coefficient Analogues Related to Lucas Sequences, arXiv:0911.3159 [math.CO], 2009.
• B. Sagan and C. Savage, Combinatorial Interpretations of Binomial Coefficient Analogues Related to Lucas Sequences, Integers 10 (2010), 697-703, A52.

Crossrefs

Columns include A000129, A084158, A099930, A099931, A383719.

Row sums are in A099928. Central column is in A099929.

Cf. A010048, A256832, A383715.

Programs

Maple

p:= proc(n) p(n):= `if`(n<2, n, 2*p(n-1)+p(n-2)) end:
f:= proc(n) f(n):= `if`(n=0, 1, p(n)*f(n-1)) end:
T:= (n, k)-> f(n)/(f(k)*f(n-k)):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Aug 15 2013

Mathematica

p[n_] := p[n] = If[n<2, n, 2*p[n-1] + p[n-2]]; f[n_] := f[n] = If[n == 0, 1, p[n] * f[n-1]]; T[n_, k_] := f[n]/(f[k]*f[n-k]); Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

Formula

P(k, n) = Prod[i=k-n+1..k, Pell(i)] / Prod[i=1..n, Pell(i)], with Pell(n) = A000129(n).

From Peter Bala, Mar 15 2013: (Start)

In terms of the Pell numbers, Pell(n) = A000129(n), the triangle entry T(n,k) = [n]!/([k]!*[n-k]!), where [n]! := Pell(1)*...*Pell(n) for n >= 1, with the convention [0]! = 1.

Define E(x) = 1 + sum {n>=0} x^n/[n]!. Then a generating function for this triangle is E(z)*E(x*z) = 1 + (1 + x)*z + (1 + 2*x + x^2)*z^2/[2]! + (1 + 5*x + 5*x^2 + x^3)*z^3/[3]! + ... (End)

G.f. of column k: x^k * exp( Sum_{j>=1} Pell((k+1)*j)/Pell(j) * x^j/j ). - Seiichi Manyama, May 07 2025