A098593 -id:A098593 - cp's OEIS frontend

A081577 Pascal-(1,2,1) array read by antidiagonals.

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 22, 10, 1, 1, 13, 46, 46, 13, 1, 1, 16, 79, 136, 79, 16, 1, 1, 19, 121, 307, 307, 121, 19, 1, 1, 22, 172, 586, 886, 586, 172, 22, 1, 1, 25, 232, 1000, 2086, 2086, 1000, 232, 25, 1, 1, 28, 301, 1576, 4258, 5944, 4258, 1576, 301, 28, 1

Offset: 0

Paul Barry, Mar 23 2003

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016777, A038764, A081583, A081584. Coefficients of the row polynomials in the Newton basis are given by A013610.
As a number triangle, this is the Riordan array (1/(1-x), x(1+2x)/(1-x)). It has row sums A002605 and diagonal sums A077947. - Paul Barry, Jan 24 2005
All entries are == 1 mod 3. - Roger L. Bagula, Oct 04 2008
Row sums are A002605. - Roger L. Bagula, Dec 09 2008
As a number triangle T, T(2n,n)=A069835(n). - Philippe Deléham, Jan 10 2014

			Square array begins as:
  1,  1,  1,   1,   1, ... A000012;
  1,  4,  7,  10,  13, ... A016777;
  1,  7, 22,  46,  79, ... A038764;
  1, 10, 46, 136, 307, ... A081583;
  1, 13, 79, 307, 886, ... A081584;
From _Roger L. Bagula_, Dec 09 2008: (Start)
As a triangle this begins:
  1;
  1,  1;
  1,  4,   1;
  1,  7,   7,    1;
  1, 10,  22,   10,    1;
  1, 13,  46,   46,   13,    1;
  1, 16,  79,  136,   79,   16,    1;
  1, 19, 121,  307,  307,  121,   19,    1;
  1, 22, 172,  586,  886,  586,  172,   22,   1;
  1, 25, 232, 1000, 2086, 2086, 1000,  232,  25,  1;
  1, 28, 301, 1576, 4258, 5944, 4258, 1576, 301, 28, 1; (End)

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Peter Bala, A note on the diagonals of a proper Riordan Array
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
Shanghua Zheng, Li Guo, and Huizhen Qiu, Extended Rota-Baxter algebras, diagonally colored Delannoy paths and Hopf algebras, arXiv:2401.11363 [math.RA], 2024. See pp. 44-45.

Cf. A002605, A081578, A081579, A081580.

Cf. Pascal-(1,a,1) array: A123562 (a=-3), A098593 (=-2), A000012 (a=-1), A007318 (a=0), A008288 (a=1), A081577(a=2), A081578 (a=3), A081579 (a=4), A081580 (a=5), A081581 (a=6), A081582 (a=7), A143683(a=8). [From Roger L. Bagula, Dec 09 2008], Philippe Deléham, Jan 10 2014, Mar 16 2014.

a081577 n k = a081577_tabl !! n !! k
a081577_row n = a081577_tabl !! n
a081577_tabl = map fst $ iterate
    (\(us, vs) -> (vs, zipWith (+) (map (* 2) ([0] ++ us ++ [0])) $
                       zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
-- Reinhard Zumkeller, Mar 16 2014

A081577:= func< n,k | (&+[Binomial(k,j)*Binomial(n-j,k)*2^j: j in [0..n-k]]) >;
[A081577(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 25 2021

a[0]={1}; a[1]={1, 1}; a[n_]:= a[n]= 2*Join[{0}, a[n-2], {0}] + Join[{0}, a[n-1]] + Join[a[n-1], {0}]; Table[a[n], {n,0,10}]//Flatten (* Roger L. Bagula, Dec 09 2008 *)
Table[Hypergeometric2F1[-k, k-n, 1, 3], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 3).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 25 2021

Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 2*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1+2*x)^k/(1-x)^(k+1).
G.f.: 1/(1-x-y-2*x*y). - Ralf Stephan, Apr 28 2004
T(n,k) = Sum_{j=0..n} binomial(k,j-k)*binomial(n+k-j,k)*2^(j-k). - Paul Barry, Oct 23 2006
a(n) = 2*{0, a(n-2), 0} + {0, a(n-1)} + {a(n-1), 0}. - Roger L. Bagula, Dec 09 2008
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 3). - Jean-François Alcover, May 24 2013
The e.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(3*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 6*x + 9*x^2/2) = 1 + 7*x + 22*x^2/2! + 46*x^3/3! + 79*x^4/4! + 121*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n,k) = A002605(n). - G. C. Greubel, May 25 2021

A099087 Expansion of 1/(1 - 2x + 2x^2).

Original entry on oeis.org

1, 2, 2, 0, -4, -8, -8, 0, 16, 32, 32, 0, -64, -128, -128, 0, 256, 512, 512, 0, -1024, -2048, -2048, 0, 4096, 8192, 8192, 0, -16384, -32768, -32768, 0, 65536, 131072, 131072, 0, -262144, -524288, -524288, 0, 1048576, 2097152, 2097152, 0, -4194304, -8388608, -8388608, 0, 16777216

Offset: 0

Paul Barry, Sep 24 2004

Yet another variation on A009545.
Row sums of Krawtchouk triangle A098593. Partial sums of e.g.f. exp(x)cos(x), or 2^(n/2)cos(Pi*n/2). See A009116.
Binomial transform of A057077. - R. J. Mathar, Nov 04 2008
Partial sums of A146559. - Philippe Deléham, Dec 01 2008
Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012
Also the inverse Catalan transform of A000079. - Arkadiusz Wesolowski, Oct 26 2012

Seiichi Manyama, Table of n, a(n) for n = 0..5000
Karl Dilcher and Maciej Ulas, Divisibility and Arithmetic Properties of a Class of Sparse Polynomials, arXiv:2008.13475 [math.NT], 2020. See Table 1, 1st column, p. 3.
Index entries for linear recurrences with constant coefficients, signature (2,-2).

Cf. A009545, A146559.

a:=[1,2];; for n in [3..50] do a[n]:=2*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Mar 16 2019

I:=[1,2]; [n le 2 select I[n] else 2*(Self(n-1) - Self(n-2)): n in [1..50]]; // G. C. Greubel, Mar 16 2019

CoefficientList[Series[1/(1 -2x +2x^2), {x, 0, 50}], x] (* Michael De Vlieger, Dec 24 2015 *)

x='x+O('x^50); Vec(1/(1-2*x+2*x^2)) \\ Altug Alkan, Dec 24 2015

[lucas_number1(n,2,2) for n in range(1, 50)] # Zerinvary Lajos, Apr 23 2009

E.g.f.: exp(x)*(cos(x) + sin(x)).
a(n) = 2^(n/2)*(cos(Pi*n/4) + sin(Pi*n/4)).
a(n) = Sum_{k=0..n} Sum_{i=0..k} binomial(n-k, k-i)*binomial(n, i) *(-1)^(k-i).
a(n) = 2*(a(n-1) - a(n-2)).
From R. J. Mathar, Apr 18 2008: (Start)
a(n) = (1-i)^(n-1) + (1+i)^(n-1) where i=sqrt(-1).
a(n) = 2 Sum_{k=0..(n-1)/2} (-1)^k*binomial(n-1,2k) if n>0. (End)
a(n) = Sum_{k=0..n} A109466(n,k)*2^k. - Philippe Deléham, Oct 28 2008
E.g.f.: (cos(x)+sin(x))*exp(x) = G(0); G(k)=1+2*x/(4*k+1-x*(4*k+1)/(2*(2*k+1)+x-2*(x^2)*(2*k+1)/((x^2)-(2*k+2)*(4*k+3)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011
G.f.: U(0) where U(k)= 1 + x*(k+3) - x*(k+1)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012
a(n) = Re((1+i)^n) + Im((1+i)^n) where i = sqrt(-1) = A146559(n) + A009545(n). - Philippe Deléham, Feb 13 2013
a(n) = Sum_{j=0..n} binomial(n, j)*(-1)^binomial(j, 2); this is the case m=2 and z=-1 of f(m,n)(z) = Sum_{j=0..n} binomial(n, j)*z^binomial(j, m). See Dilcher and Ulas. - Michel Marcus, Sep 01 2020

Signs added by N. J. A. Sloane, Nov 14 2006

A144431 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n,1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = -1.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, 2, -2, 1, 1, -3, 2, 2, -3, 1, 1, -4, 7, -8, 7, -4, 1, 1, -5, 9, -5, -5, 9, -5, 1, 1, -6, 16, -26, 30, -26, 16, -6, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, 1, -8, 29, -64, 98, -112, 98, -64, 29, -8, 1, 1, -9, 35, -75, 90, -42, -42, 90, -75, 35, -9, 1

Offset: 1

Roger L. Bagula, Oct 04 2008

Row sums are: {1, 2, 2, 0, 0, 0, 0, 0, 0, 0, ...}.
For m = ...,-1,0,1,2 we get ..., A144431, A007318 (Pascal), A008292, A060187, ..., so this might be called a sub-Pascal triangle.
The triangle starts off like A098593, but is different further on.

			Triangle begins:
  1;
  1,   1;
  1,   0,   1;
  1,  -1,  -1,   1;
  1,  -2,   2,  -2,   1;
  1,  -3,   2,   2,  -3,   1;
  1,  -4,   7,  -8,   7,  -4,   1;
  1,  -5,   9,  -5,  -5,   9,  -5,   1;
  1,  -6,  16, -26,  30, -26,  16,  -6,   1;
  1,  -7,  20, -28,  14,  14, -28,  20,  -7,   1;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 8.

Cf. A007318, A008292, A060187, A098593.

T:=proc(n,k,l) option remember;
if (n=1 or k=1 or k=n) then 1 else
(l*n-l*k+1)*T(n-1,k-1,l)+(l*k-l+1)*T(n-1,k,l); fi; end;
for n from 1 to 15 do lprint([seq(T(n,k,-1),k=1..n)]); od; # N. J. A. Sloane, May 08 2013

m=-1;
T[n_, 1]:= 1; T[n_, n_]:= 1;
T[n_, k_]:= (m*n-m*k+1)*T[n-1, k-1] + (m*k - (m - 1))*T[n-1,k];
Table[T[n, k], {n,15}, {k,n}]//Flatten

def A144431(n,k):
    if (n<3): return 1
    else: return (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3)
flatten([[A144431(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 01 2022

T(n,k) = (m*n - m*k + 1)*T(n-1, k-1) + (m*k - (m-1))*T(n-1, k) with T(n, 1) = T(n, n) = 1 and m = -1.
From G. C. Greubel, Mar 01 2022: (Start)
T(n, n-k) = T(n, k).
T(n, k) = (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3) with T(1, k) = T(2, k) = 1.
Sum_{k=1..n} T(n, k) = [n==1] + 2*[n==2] + 2*[n==3] + (1-(-1)^n)*0^(n-3)*[n>3]. (End)

Edited by N. J. A. Sloane, May 08 2013

A081578 Pascal-(1,3,1) array.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 9, 9, 1, 1, 13, 33, 13, 1, 1, 17, 73, 73, 17, 1, 1, 21, 129, 245, 129, 21, 1, 1, 25, 201, 593, 593, 201, 25, 1, 1, 29, 289, 1181, 1921, 1181, 289, 29, 1, 1, 33, 393, 2073, 4881, 4881, 2073, 393, 33, 1, 1, 37, 513, 3333, 10497, 15525, 10497, 3333, 513, 37, 1

Offset: 0

Paul Barry, Mar 23 2003

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016813, A081585, A081586. Coefficients of the row polynomials in the Newton basis are given by A013611.

As a number triangle, this is the Riordan array (1/(1-x), x*(1+3*x)/(1-x)). It has row sums A015518(n+1) and diagonal sums A103143. - Paul Barry, Jan 24 2005

			Square array begins as:
  1,  1,   1,   1,    1, ... A000012;
  1,  5,   9,  13,   17, ... A016813;
  1,  9,  33,  73,  129, ... A081585;
  1, 13,  73, 245,  593, ... A081586;
  1, 17, 129, 593, 1921, ...
As a triangle this begins:
  1;
  1,  1;
  1,  5,   1;
  1,  9,   9,    1;
  1, 13,  33,   13,     1;
  1, 17,  73,   73,    17,     1;
  1, 21, 129,  245,   129,    21,     1;
  1, 25, 201,  593,   593,   201,    25,    1;
  1, 29, 289, 1181,  1921,  1181,   289,   29,   1;
  1, 33, 393, 2073,  4881,  4881,  2073,  393,  33,  1;
  1, 37, 513, 3333, 10497, 15525, 10497, 3333, 513, 37, 1; - _Philippe Deléham_, Mar 15 2014

Vincenzo Librandi, Rows n = 0..100, flattened
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
P. Barry, A note on Krawtchouk Polynomials and Riordan Arrays, JIS 11 (2008) 08.2.2

Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).

a081578 n k = a081578_tabl !! n !! k
a081578_row n = a081578_tabl !! n
a081578_tabl = map fst $ iterate
   (\(us, vs) -> (vs, zipWith (+) (map (* 3) ([0] ++ us ++ [0])) $
                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
-- Reinhard Zumkeller, Mar 16 2014

A081578:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A081578(n,k,3): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

Table[Hypergeometric2F1[-k, k-n, 1, 4], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 4).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 3*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1+3*x)^k/(1-x)^(k+1).
T(n,k) = Sum_{j=0..n} binomial(k,j-k)*binomial(n+k-j,k)*3^(j-k). - Paul Barry, Oct 23 2006
E.g.f. for the n-th subdiagonal of the triangle, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(4*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 8*x + 16*x^2/2) = 1 + 9*x + 33*x^2/2! + 73*x^3/3! + 129*x^4/4! + 201*x^5/5! + .... - Peter Bala, Mar 05 2017
From G. C. Greubel, May 26 2021: (Start)
T(n, k, m) = Hypergeometric2F1([-k, k-n], [1], m+1), for m = 3.
T(n, k, m) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*m^j, for m = 3.
Sum_{k=0..n} T(n, k, 3) = A015518(n+1). (End)

A081582 Pascal-(1,7,1) array.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 17, 17, 1, 1, 25, 97, 25, 1, 1, 33, 241, 241, 33, 1, 1, 41, 449, 1161, 449, 41, 1, 1, 49, 721, 3297, 3297, 721, 49, 1, 1, 57, 1057, 7161, 14721, 7161, 1057, 57, 1, 1, 65, 1457, 13265, 44961, 44961, 13265, 1457, 65, 1, 1, 73, 1921, 22121, 108353, 192969, 108353, 22121, 1921, 73, 1

Offset: 0

Paul Barry, Mar 23 2003

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A017077, A081593, A081594. Coefficients of the row polynomials in the Newton basis are given by A013614.

			Rows begin
  1,  1,   1,    1,     1, ... A000012;
  1,  9,  17,   25,    33, ... A017077;
  1, 17,  97,  241,   449, ... A081593;
  1, 25, 241, 1161,  3297, ...
  1, 33, 449, 3297, 14721, ...
Triangle begins:
  1;
  1,  1;
  1,  9,    1;
  1, 17,   17,    1;
  1, 25,   97,   25,     1;
  1, 33,  241,  241,    33,    1;
  1, 41,  449, 1161,   449,   41,    1;
  1, 49,  721, 3297,  3297,  721,   49,  1;
  1, 57, 1057, 7161, 14721, 7161, 1057, 57, 1;

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A143683 (m = 8).

Cf. A015519, A017077, A081593.

A081582:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A081582(n,k,7): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

Table[ Hypergeometric2F1[-k, k-n, 1, 8], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 8).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

T(n,k) = Sum_{j = 0..n-k} binomial(n-k,j)*binomial(k,j)*8^j.
Riordan array (1/(1 - x), x*(1 + 7*x)/(1 - x)).
Square array T(n, k) defined by T(n, 0) = T(0, k)=1, T(n, k) = T(n, k-1) + 7*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1 + 7*x)^k/(1 - x)^(k+1).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 8). - Jean-François Alcover, May 24 2013
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(8*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 16*x + 64*x^2/2) = 1 + 17*x + 97*x^2/2! + 241*x^3/3! + 449*x^4/4! + 721*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n, k) = A015519(n+1). - G. C. Greubel, May 26 2021

A081579 Pascal-(1,4,1) array.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 11, 11, 1, 1, 16, 46, 16, 1, 1, 21, 106, 106, 21, 1, 1, 26, 191, 396, 191, 26, 1, 1, 31, 301, 1011, 1011, 301, 31, 1, 1, 36, 436, 2076, 3606, 2076, 436, 36, 1, 1, 41, 596, 3716, 9726, 9726, 3716, 596, 41, 1, 1, 46, 781, 6056, 21746, 33876, 21746, 6056, 781, 46, 1

Offset: 0

Paul Barry, Mar 23 2003

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016861, A081587, A081588. Coefficients of the row polynomials in the Newton basis are given by A013612.

			Square array begins as:
  1,  1,   1,    1,    1, ... A000012;
  1,  6,  11,   16,   21, ... A016861;
  1, 11,  46,  106,  191, ... A081587;
  1, 16, 106,  396, 1011, ... A081588;
  1, 21, 191, 1011, 3606, ...
As triangle this begins:
  1;
  1,  1;
  1,  6,   1;
  1, 11,  11,    1;
  1, 16,  46,   16,     1;
  1, 21, 106,  106,    21,     1;
  1, 26, 191,  396,   191,    26,     1;
  1, 31, 301, 1011,  1011,   301,    31,    1;
  1, 36, 436, 2076,  3606,  2076,   436,   36,   1;
  1, 41, 596, 3716,  9726,  9726,  3716,  596,  41,  1;
  1, 46, 781, 6056, 21746, 33876, 21746, 6056, 781, 46, 1; - _Philippe Deléham_, Mar 15 2014

Vincenzo Librandi, Rows n = 0..100, flattened
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).

Cf. A016861, A063727, A081587, A081588.

A081579:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A081579(n,k,4): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

Table[Hypergeometric2F1[-k, k-n, 1, 5], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 5).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 4*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1+4*x)^k/(1-x)^(k+1).
From Philippe Deléham, Mar 15 2014: (Start)
Riordan array (1/(1-x), x*(1+4*x)/(1-x)).
Sum_{k=0..n} T(n, k) = A063727(n). (End)
E.g.f. for the n-th subdiagonal of the triangle, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(5*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 10*x + 25*x^2/2) = 1 + 11*x + 46*x^2/2! + 106*x^3/3! + 191*x^4/4! + 301*x^5/5! + .... - Peter Bala, Mar 05 2017
From G. C. Greubel, May 26 2021: (Start)
T(n, k, m) = Hypergeometric2F1([-k, k-n], [1], m+1), for m = 4.
T(n, k, m) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*m^j, for m = 4. (End)

A081580 Pascal-(1,5,1) array.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 13, 13, 1, 1, 19, 61, 19, 1, 1, 25, 145, 145, 25, 1, 1, 31, 265, 595, 265, 31, 1, 1, 37, 421, 1585, 1585, 421, 37, 1, 1, 43, 613, 3331, 6145, 3331, 613, 43, 1, 1, 49, 841, 6049, 17401, 17401, 6049, 841, 49, 1, 1, 55, 1105, 9955, 40105, 65527, 40105, 9955, 1105, 55, 1

Offset: 0

Paul Barry, Mar 23 2003

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016921, A081589, A081590. Coefficients of the row polynomials in the Newton basis are given by A013613.

			Square array begins as:
  1,  1,   1,    1,    1, ... A000012;
  1,  7,  13,   19,   25, ... A016921;
  1, 13,  61,  145,  265, ... A081589;
  1, 19, 145,  595, 1585, ... A081590;
  1, 25, 265, 1585, 6145, ...
The triangle begins as:
  1;
  1,  1;
  1,  7,    1;
  1, 13,   13,    1;
  1, 19,   61,   19,     1;
  1, 25,  145,  145,    25,     1;
  1, 31,  265,  595,   265,    31,     1;
  1, 37,  421, 1585,  1585,   421,    37,    1;
  1, 43,  613, 3331,  6145,  3331,   613,   43,    1;
  1, 49,  841, 6049, 17401, 17401,  6049,  841,   49,  1;
  1, 55, 1105, 9955, 40105, 65527, 40105, 9955, 1105, 55, 1; - _Philippe Deléham_, Mar 15 2014

Vincenzo Librandi, Rows n = 0..100, flattened
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).

Cf. A002532, A016921, A081589, A081590.

A081580:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A081580(n,k,5): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

Table[Hypergeometric2F1[-k, k-n, 1, 6], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 6).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 5*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1+5*x)^k/(1-x)^(k+1).
From Paul Barry, Aug 28 2008: (Start)
Number triangle T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*5^j.
Riordan array (1/(1-x), x*(1+5*x)/(1-x)). (End)
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 6). - Jean-François Alcover, May 24 2013
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(6*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 12*x + 36*x^2/2) = 1 + 13*x + 61*x^2/2! + 145*x^3/3! + 265*x^4/4! + 421*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n, k, 3) = A002532(n+1). - G. C. Greubel, May 26 2021

A081581 Pascal-(1,6,1) array.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 15, 15, 1, 1, 22, 78, 22, 1, 1, 29, 190, 190, 29, 1, 1, 36, 351, 848, 351, 36, 1, 1, 43, 561, 2339, 2339, 561, 43, 1, 1, 50, 820, 5006, 9766, 5006, 820, 50, 1, 1, 57, 1128, 9192, 28806, 28806, 9192, 1128, 57, 1, 1, 64, 1485, 15240, 68034, 116208, 68034, 15240, 1485, 64, 1

Offset: 0

Paul Barry, Mar 23 2003

One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016993, A081591, A081592. Coefficients of the row polynomials in the Newton basis are given by A013614.

			Rows start as:
  1,  1,   1,    1,    1, ... A000012;
  1,  8,  15,   22,   29, ... A016993;
  1, 15,  78,  190,  351, ... A081591;
  1, 22, 190,  848, 2339, ...
  1, 29, 351, 2339, 9766, ...
The triangle starts as:
  1;
  1,  1;
  1,  8,   1;
  1, 15,  15,    1;
  1, 22,  78,   22,    1;
  1, 29, 190,  190,   29,   1;
  1, 36, 351,  848,  351,  36,  1;
  1, 43, 561, 2339, 2339, 561, 43, 1;

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081582 (m = 7), A143683 (m = 8).

Cf. A016993, A081591, A083099.

A081581:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A081581(n,k,6): k in [0..n], n in [0..12]]; // G. C. Greubel, May 26 2021

Table[Hypergeometric2F1[-k, k-n, 1, 7], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

t(n, k) = sum(j=0, n-k, binomial(n-k, j)*binomial(k, j)*7^j) \\ Michel Marcus, May 24 2013

flatten([[hypergeometric([-k, k-n], [1], 7).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 26 2021

Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 6*T(n-1, k-1) + T(n-1, k).
Rows are the expansions of (1+6*x)^k/(1-x)^(k+1).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 7). - Jean-François Alcover, May 24 2013
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(7*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 14*x + 49*x^2/2) = 1 + 15*x + 78*x^2/2! + 190*x^3/3! + 351*x^4/4! + 561*x^5/5! + .... - Peter Bala, Mar 05 2017
From G. C. Greubel, May 26 2021: (Start)
T(n, k, m) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*m^j, for m = 6.
Sum_{k=0..n} T(n, k, 6) = A083099(n+1). (End)

A144432 Triangle, T(n, k), read by rows: T(n, k) = t(n, k)^2 - t(n, k) - 1, where t(n,k) = (m(n-k) + 1)t(n-1, k-1) + (mk - (m-1))t(n-1, k) and m = -1.

Original entry on oeis.org

-1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, 5, 1, 5, -1, -1, 11, 1, 1, 11, -1, -1, 19, 41, 71, 41, 19, -1, -1, 29, 71, 29, 29, 71, 29, -1, -1, 41, 239, 701, 869, 701, 239, 41, -1, -1, 55, 379, 811, 181, 181, 811, 379, 55, -1

Offset: 1

Roger L. Bagula, Oct 04 2008

			Triangle begins as:
  -1;
  -1, -1;
  -1, -1,  -1;
  -1,  1,   1,  -1;
  -1,  5,   1,   5,  -1;
  -1, 11,   1,   1,  11,  -1;
  -1, 19,  41,  71,  41,  19,  -1;
  -1, 29,  71,  29,  29,  71,  29,  -1;
  -1, 41, 239, 701, 869, 701, 239,  41, -1;
  -1, 55, 379, 811, 181, 181, 811, 379, 55, -1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A098593, A144431.

t[n_, k_, m_]:= t[n, k, m]= If[k==1 || k==n, 1, (m*(n-k)+1)*t[n-1,k-1,m] + (m*(k - 1)+1)*t[n-1,k,m]];
T[n_, k_, m_]:= t[n,k,m]^2 -t[n,k,m] -1;
Table[T[n,k,-1], {n,15}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 02 2022 *)

def t(n,k):
    if (n<3): return 1
    else: return (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3)
def A144432(n,k): return t(n,k)^2 - t(n,k) - 1
flatten([[A144432(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 02 2022

T(n, k) = t(n, k)^2 - t(n, k) - 1, where t(n,k) = (m*(n-k) + 1)*t(n-1, k-1) + (m*k - (m-1))*t(n-1, k) and m = -1.
From G. C. Greubel, Mar 02 2022: (Start)
T(n, n-k) = T(n, k).
T(n, k) = t(n,k)^2 - t(n,k) - 1, where t(n,k) = (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3) with t(1, k) = t(2, k) = 1.
Sum_{k=1..n} T(n,k) = -n*[n<4] + ( 2*binomial(2*n-6, n-3)*(binomial(n-1,2) - (-1)^n*binomial(n-3,2))/binomial(n-1,2) - n )*[n>=4]. (End)

Edited by G. C. Greubel, Mar 02 2022

A123562 Pascal-(1,-3,1) array, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, -3, -3, 1, 1, -5, -3, -5, 1, 1, -7, 1, 1, -7, 1, 1, -9, 9, 11, 9, -9, 1, 1, -11, 21, 17, 17, 21, -11, 1, 1, -13, 37, 11, 1, 11, 37, -13, 1, 1, -15, 57, -15, -39, -39, -15, 57, -15, 1, 1, -17, 81, -69, -87, -81, -87, -69, 81, -17, 1

Offset: 0

Philippe Deléham, Nov 12 2006

Riordan array (1/(1-x), x*(1-3x)/(1-x)).

			Triangle begins:
  1;
  1,   1;
  1,  -1,   1;
  1,  -3,  -3,   1;
  1,  -5,  -3,  -5,   1;
  1,  -7,   1,   1,  -7,   1;
  1,  -9,   9,  11,   9,  -9,  1;
  1, -11,  21,  17,  17,  21, -11,   1;
  1, -13,  37,  11,   1,  11,  37, -13,   1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. Pascal (1,m,1) array: A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).

T[n_, k_] := Sum[Binomial[n - j, k]*Binomial[k, j]*(-3)^j, {j, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 15 2017 *)

for(n=0,10, for(k=0,n, print1(sum(j=0,n, binomial(n-j,k)*  binomial(k,j)*(-3)^j), ", "))) \\ G. C. Greubel, Oct 15 2017

Sum_{k=0..n} T(n,k) = A088137(n+1).
T(n,k) = T(n-1,k-1) + T(n-1,k) - 3*T(n-2,k-1), n>0.
From Paul Barry, Jan 24 2011: (Start)
T(n,k) = Sum_{j=0..n} binomial(n-j,k)*binomial(k,j)*(-3)^j.
T(n,k) = [k<=n]*Hypergeometric2F1(-k,k-n,1,-2). (End)
E.g.f. for the n-th subdiagonal: exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(-2*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 - 4*x + 4*x^2/2) = 1 - 3*x - 3*x^2/2! + x^3/3! + 9*x^4/4! + 21*x^5/5! + .... - Peter Bala, Mar 05 2017

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A081577 Pascal-(1,2,1) array read by antidiagonals.

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A099087 Expansion of 1/(1 - 2*x + 2*x^2).

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A144431 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n,1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = -1.

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A081578 Pascal-(1,3,1) array.

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A081582 Pascal-(1,7,1) array.

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A099087 Expansion of 1/(1 - 2x + 2x^2).

A144431 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n,1) = T(n,n) = 1, otherwise T(n, k) = (mn-mk+1)T(n-1,k-1) + (mk-m+1)*T(n-1,k), where m = -1.

A144432 Triangle, T(n, k), read by rows: T(n, k) = t(n, k)^2 - t(n, k) - 1, where t(n,k) = (m(n-k) + 1)t(n-1, k-1) + (mk - (m-1))t(n-1, k) and m = -1.