A099929 -id:A099929 - cp's OEIS frontend

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

Ralf Stephan, Nov 03 2004

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

			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;
  ...

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.

Columns include A000129, A084158, A099930, A099931, A383719.
Row sums are in A099928. Central column is in A099929.
Cf. A010048, A256832, A383715.

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

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 *)

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

A256831 Decimal expansion of Pell factorial constant.

Original entry on oeis.org

1, 1, 4, 1, 9, 8, 2, 5, 6, 9, 6, 6, 7, 7, 9, 1, 2, 0, 6, 0, 2, 8, 0, 4, 3, 3, 3, 8, 3, 6, 7, 8, 6, 0, 1, 5, 0, 8, 6, 4, 7, 3, 0, 4, 8, 2, 4, 0, 8, 5, 4, 0, 7, 9, 1, 5, 5, 6, 2, 5, 4, 3, 5, 2, 4, 4, 9, 8, 4, 3, 7, 8, 5, 4, 8, 0, 6, 2, 0, 8, 6, 0, 7, 8, 2, 5, 0, 6, 3, 7, 0, 6, 0, 9, 2, 5, 3, 3, 4, 7, 8, 1, 6, 3, 6

Offset: 1

Vaclav Kotesovec, Apr 10 2015

			1.141982569667791206028043338367860150864730482408540791556...

Cf. A000129, A099929, A256832.

Cf. A062073, A218490, A253924.

RealDigits[N[QPochhammer[2*Sqrt[2]-3], 105]][[1]]

Equals limit n->infinity A256832(n) / ((1+sqrt(2))^(n*(n+1)/2) / 2^(3*n/2)).

A256799 Catalan number analogs for A099927, the generalized binomial coefficients for Pell numbers (A000129).

Original entry on oeis.org

1, 1, 6, 203, 40222, 46410442, 312163223724, 12237378320283699, 2796071362211148193590, 3723566980632561787914135870, 28901575272390972687956930234335380, 1307480498356321410289575304307661963042110, 344746842780849469098742541704318199701366091840620

Offset: 0

Tom Edgar, Apr 10 2015

nonn

One definition of the Catalan numbers is binomial(2*n,n) / (n+1); the current sequence models this definition using the generalized binomial coefficients arising from Pell numbers (A000129).

			a(5) = Pell(10)..Pell(7)/Pell(5)..Pell(1) = (2378*985*408*169)/(29*12*5*2*1) = 46410442.
a(3) = A099927(6,3)/Pell(3) = 2436/12 = 203.

Alois P. Heinz, Table of n, a(n) for n = 0..50
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.

Cf. A000129, A099927, A003150, A099929, A256831, A256832.

p:= n-> (<<2|1>, <1|0>>^n)[1, 2]:
a:= n-> mul(p(i), i=n+2..2*n)/mul(p(i), i=1..n):
seq(a(n), n=0..12);  # Alois P. Heinz, Apr 10 2015

Pell[m_]:=Expand[((1+Sqrt[2])^m-(1-Sqrt[2])^m)/(2*Sqrt[2])]; Table[Product[Pell[k],{k,1,2*n}]/(Product[Pell[k],{k,1,n}])^2 / Pell[n+1],{n,0,15}] (* Vaclav Kotesovec, Apr 10 2015 *)

P=[lucas_number1(n, 2, -1) for n in [0..30]]
[1/P[n+1]*prod(P[1:2*n+1])/(prod(P[1:n+1]))^2 for n in [0..14]]

a(n) = Pell(2n)Pell(2n-1)...Pell(n+2)/Pell(n)Pell(n-1)...Pell(1) = A099927(2*n,n)/Pell(n+1) = A099929(n)/Pell(n+1), where Pell(k) = A000129(k).

a(n) ~ 2^(3/2) * (1+sqrt(2))^(n^2-n-1) / c, where c = A256831 = 1.141982569667791206028... . - Vaclav Kotesovec, Apr 10 2015

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

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A256831 Decimal expansion of Pell factorial constant.

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A256799 Catalan number analogs for A099927, the generalized binomial coefficients for Pell numbers (A000129).

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