cp's OEIS Frontend

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

%I A099927 #36 May 07 2025 11:13:47
%S A099927 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,
%T A099927 1015,70,1,1,169,5915,34307,34307,5915,169,1,1,408,34476,482664,
%U A099927 1166438,482664,34476,408,1,1,985,200940,6791772,39618670,39618670,6791772,200940,985,1
%N A099927 Pellonomial triangle P(k,n) read by rows.
%C A099927 Also (signed) coefficients of solutions to 0 = Sum[i=0..k+1, x(i)*Pell(m+i)^k ].
%C A099927 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
%H A099927 Alois P. Heinz, <a href="/A099927/b099927.txt">Rows n = 0..56, flattened</a>
%H A099927 Tom Edgar and Michael Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Edgar/edgar3.html">Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
%H A099927 S. Falcon, <a href="http://saspublisher.com/wp-content/uploads/2014/06/SJET24C669-675.pdf">On The Generating Functions of the Powers of the K-Fibonacci Numbers</a>, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
%H A099927 B. Sagan and C. Savage, <a href="http://arxiv.org/abs/0911.3159">Combinatorial Interpretations of Binomial Coefficient Analogues Related to Lucas Sequences</a>, arXiv:0911.3159 [math.CO], 2009.
%H A099927 B. Sagan and C. Savage, <a href="http://www.emis.de/journals/INTEGERS/papers/k52/k52.Abstract.html">Combinatorial Interpretations of Binomial Coefficient Analogues Related to Lucas Sequences</a>, Integers 10 (2010), 697-703, A52.
%F A099927 P(k, n) = Prod[i=k-n+1..k, Pell(i)] / Prod[i=1..n, Pell(i)], with Pell(n) = A000129(n).
%F A099927 From _Peter Bala_, Mar 15 2013: (Start)
%F A099927 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.
%F A099927 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)
%F A099927 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
%e A099927 Triangle starts:
%e A099927   1;
%e A099927   1,   1;
%e A099927   1,   2,    1;
%e A099927   1,   5,    5,     1;
%e A099927   1,  12,   30,    12,     1;
%e A099927   1,  29,  174,   174,    29,    1;
%e A099927   1,  70, 1015,  2436,  1015,   70,   1;
%e A099927   1, 169, 5915, 34307, 34307, 5915, 169, 1;
%e A099927   ...
%p A099927 p:= proc(n) p(n):= `if`(n<2, n, 2*p(n-1)+p(n-2)) end:
%p A099927 f:= proc(n) f(n):= `if`(n=0, 1, p(n)*f(n-1)) end:
%p A099927 T:= (n, k)-> f(n)/(f(k)*f(n-k)):
%p A099927 seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Aug 15 2013
%t A099927 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_ *)
%Y A099927 Columns include A000129, A084158, A099930, A099931, A383719.
%Y A099927 Row sums are in A099928. Central column is in A099929.
%Y A099927 Cf. A010048, A256832, A383715.
%K A099927 nonn,tabl
%O A099927 0,5
%A A099927 _Ralf Stephan_, Nov 03 2004