A122433 -id:A122433 - cp's OEIS frontend

A159854 Riordan array (1-x,x/(1-x)).

1, -1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 3, 1, 0, 0, 1, 4, 6, 4, 1, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 0, 1, 8, 28, 56, 70, 56, 28, 8, 1, 0, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1

Offset: 0

Philippe Deléham, Apr 24 2009

From Peter Bala, Sep 13 2015: (Start)
The m-th power of the array is the Riordan array (1 - m*x, x/(1 - m*x)).
This array, call it M, is a pseudo-involution in the Riordan group, that is, M*D has order 2, where D = (1,-z) is the diagonal matrix with alternating 1's and -1's on the main diagonal.
This array belongs to the subgroups G := { (f(x)/(x*f'(x)),f(x)): f(x) = x + c(2)*x^2 + c(3)*x^3 + ..., c(i) integral } and H := { (x/f(x),f(x)): f(x) = x + c(2)*x^2 + c(3)*x^3 + ..., c(i) integral } of the Riordan group. Moreover, this array generates the infinite cyclic group (G intersect H). Compare with Pascal's triangle (A007318) which generates the intersection of the Bell subgroup and the hitting-time subgroup of the Riordan group.
(End)

			Triangle begins:
   1
  -1,1
   0,0,1
   0,0,1,1
   0,0,1,2,1
   0,0,1,3,3,1
...

Muniru A Asiru, Table of n, a(n) for n = 0..5151
Igor Victorovich Statsenko, On the ordinal numbers of triangles of generalized special numbers, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.

Cf. A007318, A122433, A159855.

Cf. A144225. - R. J. Mathar, Oct 24 2009

Flat(List([0..12],n->List([0..n],k->Binomial(n,k)-2*Binomial(n-1,n-k-1)+Binomial(n-2,n-k-2)))); # Muniru A Asiru, Mar 22 2018

/* As triangle */ [[Binomial(n,n-k)-2*Binomial(n-1,n-k-1)+ Binomial(n-2,n-k-2): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 11 2019

seq(seq( binomial(n-2,k-2), k = 0..n), n = 0..12); # Peter Bala, Mar 20 2018

Table[Binomial[n-2, k-2], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 11 2019 *)

# uses[riordan_array from A256893]
riordan_array(1-x, x/(1-x), 8) # Peter Luschny, Mar 21 2018

Exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(x^2/2! + x^3/3!) = x^2/2! + 4*x^3/3! + 10*x^4/4! + 20*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014
T(n,k) = C(n,n-k) - 2*C(n-1,n-k-1) + C(n-2,n-k-2), where C(n,k) = n!/(k!*(n-k)!) for 0 <= k <= n, otherwise 0. Cf. A159855. - Peter Bala, Mar 20 2018
T(n,k) = Sum_{i=0..n-k} binomial(n+1, n-k-i)*Stirling2(i + m + 1, i+1) *(-1)^i, where m = 1 for n >= 0, 0 <= k <= n. See A007318, A370516 for m=0 and m=2. - Igor Victorovich Statsenko, Feb 26 2023

A122434 Expansion of (1+x)^3/(1+x+x^2).

Original entry on oeis.org

1, 2, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0

Offset: 0

Paul Barry, Sep 04 2006

Row sums of number triangle A122433. Binomial transform is A077859.

Index entries for linear recurrences with constant coefficients, signature (-1,-1).

CoefficientList[Series[(1+x)^3/(1+x+x^2),{x,0,70}],x] (* or *) PadRight[ {1,2,0},71,{-1,1,0}] (* Harvey P. Dale, Apr 25 2012 *)

a(n) = 2*sqrt(3)*cos(2*Pi*n/3+Pi/6)/3+C(1,n)+C(0,n).

a(n) = -A049347(n) if n>=2 . - R. J. Mathar, Feb 08 2008

A375550 Triangle read by rows: T(m, n, k) = binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1) for m = 4.

Original entry on oeis.org

1, 6, 1, 25, 7, 1, 88, 32, 8, 1, 280, 120, 40, 9, 1, 832, 400, 160, 49, 10, 1, 2352, 1232, 560, 209, 59, 11, 1, 6400, 3584, 1792, 769, 268, 70, 12, 1, 16896, 9984, 5376, 2561, 1037, 338, 82, 13, 1, 43520, 26880, 15360, 7937, 3598, 1375, 420, 95, 14, 1

Offset: 0

Peter Luschny, Sep 23 2024

Triangle T(m,n,k) is a Riordan array of the form ((1-x)^(m-1)*(1-2x)^(-m-1), x/(1-x)), for m = 3. - Igor Victorovich Statsenko, Feb 08 2025

			Triangle starts:
  [0]     1;
  [1]     6,     1;
  [2]    25,     7,     1;
  [3]    88,    32,     8,    1;
  [4]   280,   120,    40,    9,    1;
  [5]   832,   400,   160,   49,   10,    1;
  [6]  2352,  1232,   560,  209,   59,   11,   1;
  [7]  6400,  3584,  1792,  769,  268,   70,  12,  1;
  [8] 16896,  9984,  5376, 2561, 1037,  338,  82, 13,  1;
  [9] 43520, 26880, 15360, 7937, 3598, 1375, 420, 95, 14, 1;
  ...
Seen as an array of the columns:
  [0] 1,  6, 25,  88,  280,  832,  2352,  6400,  16896, ...
  [1] 1,  7, 32, 120,  400, 1232,  3584,  9984,  26880, ...
  [2] 1,  8, 40, 160,  560, 1792,  5376, 15360,  42240, ...
  [3] 1,  9, 49, 209,  769, 2561,  7937, 23297,  65537, ...
  [4] 1, 10, 59, 268, 1037, 3598, 11535, 34832, 100369, ...
  [5] 1, 11, 70, 338, 1375, 4973, 16508, 51340, 151709, ...
  [6] 1, 12, 82, 420, 1795, 6768, 23276, 74616, 226325, ...

Column k: A055585 (k=0), A001794 (k=1), A001789 (k=2), A027608 (k=3), A055586 (k=4).

Cf. A145018 (diagonal n-2), A375549 (row sums), A049612 (alternating row sums), A122433.

T := (m, n, k) -> binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1);
for n from 0 to 9 do seq(simplify(T(4, n, k)), k = 0..n) od;
# As a binomial sum:
T := (m, n, k) -> add(binomial(m + j, m)*binomial(n + 1, n - (j + k)), j = 0..n-k):
for n from 0 to 9 do [n], seq(T(3, n, k), k = 0..n) od;
# Alternative, generating the array of the columns:
cgf := k -> (1 - x)^(2 - k) / (1 - 2*x)^4:
ser := (k, len) -> series(cgf(k), x, len + 2):
Tcol := (k, len) -> seq(coeff(ser(k, len), x, j), j = 0..len):
seq(lprint([k], Tcol(k, 8)), k = 0..6);

T(m, n, k) = Sum_{j=0..n-k} binomial(m + j, m)*binomial(n + 1, n - (j + k)) for m = 3.

G.f. of column k: (1 - x)^(2 - k) / (1 - 2*x)^4.

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A159854 Riordan array (1-x,x/(1-x)).

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A122434 Expansion of (1+x)^3/(1+x+x^2).

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A375550 Triangle read by rows: T(m, n, k) = binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1) for m = 4.

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