A254749 -id:A254749 - cp's OEIS frontend

A123021 Triangle of coefficients of (1 - x)^n*B(x/(1 - x),n), where B(x,n) is the Morgan-Voyce polynomial related to A078812.

1, 2, -1, 3, -2, 4, -2, -2, 1, 5, 0, -9, 6, -1, 6, 5, -24, 18, -4, 7, 14, -49, 36, -4, -4, 1, 8, 28, -84, 50, 20, -30, 10, -1, 9, 48, -126, 36, 115, -120, 45, -6, 10, 75, -168, -48, 358, -335, 120, -6, -6, 1, 11, 110, -198, -264, 847, -714, 175, 84, -63, 14

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 24 2006

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A078812(n,j)*x^j*(1 - x)^(n - j).

			Triangle begins:
    1;
    2,  -1;
    3,  -2;
    4,  -2,   -2,    1;
    5,   0,   -9,    6,  -1;
    6,   5,  -24,   18,  -4;
    7,  14,  -49,   36,  -4,   -4,   1;
    8,  28,  -84,   50,  20,  -30,  10, -1;
    9,  48, -126,   36, 115, -120,  45, -6;
   10,  75, -168,  -48, 358, -335, 120, -6,  -6,  1;
   11, 110, -198, -264, 847, -714, 175, 84, -63, 14, -1;
   ... - _Franck Maminirina Ramaharo_, Oct 09 2018

G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened
Thomas Koshy, Morgan-Voyce Polynomials, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, 2001, pp. 480-495.
M. N. S. Swamy, Rising Diagonal Polynomials Associated with Morgan-Voyce Polynomials, The Fibonacci Quarterly Vol. 38 (2000), 61-70.
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials

Cf. A078812, A085478.

Cf. A122753, A123018, A123019, A123027, A123199, A123202, A123217, A123221.

Table[CoefficientList[Sum[Binomial[n+k+1, n-k]*x^k*(1-x)^(n-k), {k, 0, n}], x], {n, 0, 10}]//Flatten

t(n, k) := binomial(n + k + 1, n - k)$
P(x, n) := expand(sum(t(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
T(n, k) := ratcoef(P(x, n), x, k)$
tabf(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, hipow(P(x, n), x)))$ /* Franck Maminirina Ramaharo, Oct 09 2018 */

def p(n,x): return sum( binomial(n+j+1, n-j)*x^j*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (0..12)]) # G. C. Greubel, Jul 15 2021

From Franck Maminirina Ramaharo, Oct 09 2018: (Start)
Row n = coefficients in the expansion of (1/sqrt((4 - 3*x)*x))*(((2 - x + sqrt((4 - 3*x)*x))/2)^(n + 1) - ((2 - x - sqrt((4 - 3*x)*x))/2)^(n + 1)).
G.f.: 1/(1 - (2 - x)*y + (1 - x)^2*y^2).
E.g.f.: (1/sqrt((4 - 3*x)*x))*((2 - x + sqrt((4 - 3*x)*x))*exp(y*(2 - x + sqrt((4 - 3*x)*x))/2)/2 - (2 - x - sqrt((4 - 3*x)*x))*exp(y*(2 - x - sqrt((4 - 3*x)*x))/2)/2).
T(n,1) = -A254749(n+1). (End)

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 09 2018

A112466 Riordan array ((1+2*x)/(1+x), x/(1+x)).

Original entry on oeis.org

1, 1, 1, -1, 0, 1, 1, -1, -1, 1, -1, 2, 0, -2, 1, 1, -3, 2, 2, -3, 1, -1, 4, -5, 0, 5, -4, 1, 1, -5, 9, -5, -5, 9, -5, 1, -1, 6, -14, 14, 0, -14, 14, -6, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, -1, 8, -27, 48, -42, 0, 42, -48, 27, -8, 1, 1, -9, 35, -75, 90, -42, -42, 90, -75, 35, -9, 1, -1, 10, -44, 110, -165, 132, 0, -132, 165, -110, 44, -10, 1

Offset: 0

Paul Barry, Sep 06 2005

Inverse is A112465.
Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 07 2006; corrected by Philippe Deléham, Dec 11 2008
Equals A097808 when the first column is removed. - Georg Fischer, Jul 26 2023

			Triangle starts
   1;
   1,  1;
  -1,  0,  1;
   1, -1, -1,  1;
  -1,  2,  0, -2,  1;
   1, -3,  2,  2, -3,  1;
  -1,  4, -5,  0,  5, -4,  1;
From _Paul Barry_, Apr 08 2011: (Start)
Production matrix begins
   1,  1;
  -2, -1,  1;
   2,  0, -1,  1;
  -2,  0,  0, -1,  1;
   2,  0,  0,  0, -1,  1;
  -2,  0,  0,  0,  0, -1,  1;
   2,  0,  0,  0,  0,  0, -1,  1; (End)

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.

Cf. A000007, A008482, A037012, A057079, A097808, A112467.

Columns: A248157(n+2) (k=1), (-1)^n*A080956(n-2) (k=2), (-1)^(n-1)*A254749(n-2) (k=3).

A112466:= func< n,k | n eq 0 select 1 else (-1)^(n+k)*(Binomial(n,k) - 2*Binomial(n-1,k)) >;
[A112466(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 30 2025

seq(seq( (-1)^(n-k)*(2*binomial(n-1, k-1)-binomial(n, k)), k=0..n), n=0..10); # G. C. Greubel, Feb 19 2020

{1}~Join~Table[(Binomial[n, n - k] - 2 Binomial[n - 1, n - k - 1])*(-1)^(n - k), {n, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 18 2020 *)

T(n,k) = (-1)^(n-k)*(binomial(n, n-k) - 2*binomial(n-1, n-k-1)); \\ Michel Marcus, Feb 19 2020

def A112466(n,k): return 1 if (n==0) else (-1)^(n+k)*(binomial(n,k) - 2*binomial(n-1,k))
print(flatten([[A112466(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 30 2025

Number triangle: T(n,k) = (-1)^(n-k)*(C(n, n-k) - 2*C(n-1, n-k-1)), with T(0,0) = 1.
T(2*n, n) = 0 (main diagonal).
Sum_{k=0..n} T(n, k) = 0 + [n=0] + 2*[n=1] (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^(n+1)*Fibonacci(n-2) (diagonal sums).
Sum_{k=0..n} T(n,k)*x^k = (x+1)*(x-1)^(n-1), for n >= 1. - Philippe Deléham, Oct 03 2005
T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if n < 0 or if n < k, T(n,k) = T(n-1,k-1) - T(n-1,k) for n > 1. - Philippe Deléham, Nov 26 2006
G.f.: (1+2*x)/(1+x-x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Apr 30 2025: (Start)
T(2*n+1, 2*n+1-k) = T(2*n+1, k) (symmetric odd n rows).
T(2*n, 2*n-k) = (-1)*T(2*n, k) (antisymmetric even n rows).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n) (signed row sums).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^n*A057079(n+2) (signed diagonal sums). (End)

cp's OEIS Frontend

A123021 Triangle of coefficients of (1 - x)^n*B(x/(1 - x),n), where B(x,n) is the Morgan-Voyce polynomial related to A078812.

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