A081580 -id:A081580 - cp's OEIS frontend

A081589 Third row of Pascal-(1,5,1) array A081580.

1, 13, 61, 145, 265, 421, 613, 841, 1105, 1405, 1741, 2113, 2521, 2965, 3445, 3961, 4513, 5101, 5725, 6385, 7081, 7813, 8581, 9385, 10225, 11101, 12013, 12961, 13945, 14965, 16021, 17113, 18241, 19405, 20605, 21841, 23113, 24421, 25765, 27145, 28561, 30013, 31501

Offset: 0

Paul Barry, Mar 23 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016921, A081580, A081590.

[1-6*n+18*n^2: n in [0..40]]; // Vincenzo Librandi, Aug 09 2013

CoefficientList[Series[(1 + 5 x)^2 / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 09 2013 *)

a(n)=18*n^2-6*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 1 - 6*n + 18*n^2.
G.f.: (1+5*x)^2/(1-x)^3.
From Elmo R. Oliveira, Jun 06 2025: (Start)
E.g.f.: exp(x)*(1 + 12*x + 18*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)

A081590 Fourth row of Pascal-(1,5,1) array A081580.

Original entry on oeis.org

1, 19, 145, 595, 1585, 3331, 6049, 9955, 15265, 22195, 30961, 41779, 54865, 70435, 88705, 109891, 134209, 161875, 193105, 228115, 267121, 310339, 357985, 410275, 467425, 529651, 597169, 670195, 748945, 833635, 924481, 1021699, 1125505, 1236115, 1353745, 1478611

Offset: 0

Paul Barry, Mar 23 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081580, A081589.

[1+36*n-54*n^2+36*n^3: n in [0..40]]; // Vincenzo Librandi, Sep 07 2011

Table[36n^3-54n^2+36n+1,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,19,145,595},50] (* Harvey P. Dale, Sep 07 2011 *)

a(n) = 1 + 36*n - 54*n^2 + 36*n^3.
G.f.: (1+5*x)^3/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=19, a(2)=145, a(3)=595. - Harvey P. Dale, Sep 07 2011
E.g.f.: exp(x)*(1 + 18*x + 54*x^2 + 36*x^3). - Elmo R. Oliveira, Jun 06 2025

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

Original entry on oeis.org

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

A098593 A triangle of Krawtchouk coefficients.

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, -1, 0, -1, -4, 1, 1, -5, 1, 3, 3, 1, -5, 1, 1, -6, 4, 6, 6, 6, 4, -6, 1, 1, -7, 8, 8, 6, 6, 8, 8, -7, 1, 1, -8, 13, 8, 2, 0, 2, 8, 13, -8, 1, 1, -9, 19, 5, -6, -10, -10, -6, 5, 19, -9, 1, 1, -10, 26, -2, -17, -20, -20, -20, -17, -2, 26, -10, 1, 1, -11, 34, -14, -29, -25

Offset: 0

Paul Barry, Sep 17 2004

Row sums are A009545(n+1), with e.g.f. exp(x)(cos(x)+sin(x)). Diagonal sums are A077948.
The rows are the diagonals of the Krawtchouk matrices. Coincides with the Riordan array (1/(1-x),(1-2x)/(1-x)). - Paul Barry, Sep 24 2004
Corresponds to Pascal-(1,-2,1) array, read by antidiagonals. The Pascal-(1,-2,1) array has n-th row generated by (1-2x)^n/(1-x)^(n+1). - Paul Barry, Sep 24 2004
A modified version (different signs) of this triangle is given by T(n,k) = Sum_{j=0..n} C(n-k,j)*C(k,j)*cos(Pi*(k-j)). - Paul Barry, Jun 14 2007

			Rows begin {1}, {1,1}, {1,0,1}, {1,-1,-1,1}, {1,-2,-2,-2,1}, ...
From _Paul Barry_, Oct 05 2010: (Start)
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, -1,  0, -1, -4,  1,
  1, -5,  1,  3,  3,  1, -5,  1,
  1, -6,  4,  6,  6,  6,  4, -6,  1,
  1, -7,  8,  8,  6,  6,  8,  8, -7,  1,
  1, -8, 13,  8,  2,  0,  2,  8, 13, -8,  1
Production matrix (related to large Schroeder numbers A006318) begins
  1,     1,
  0,    -1,     1,
  0,    -2,    -1,    1,
  0,    -6,    -2,   -1,   1,
  0,   -22,    -6,   -2,  -1,   1,
  0,   -90,   -22,   -6,  -2,  -1,  1,
  0,  -394,   -90,  -22,  -6,  -2, -1,  1,
  0, -1806,  -394,  -90, -22,  -6, -2, -1,  1,
  0, -8558, -1806, -394, -90, -22, -6, -2, -1, 1
Production matrix of inverse is
    -1,   1,
    -2,   1,  1,
    -4,   2,  1,  1,
    -8,   4,  2,  1,  1,
   -16,   8,  4,  2,  1, 1,
   -32,  16,  8,  4,  2, 1, 1,
   -64,  32, 16,  8,  4, 2, 1, 1,
  -128,  64, 32, 16,  8, 4, 2, 1, 1,
  -256, 128, 64, 32, 16, 8, 4, 2, 1, 1 (End)

P. Feinsilver and J. Kocik, Krawtchouk matrices from classical and quantum walks, Contemporary Mathematics, 287 2001, pp. 83-96.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Paul Barry, A note on Krawtchouk Polynomials and Riordan Arrays, JIS 11 (2008) 08.2.2
P. Feinsilver and J. Kocik, Krawtchouk polynomials and Krawtchouk matrices, arxiv:quant-ph/0702073, 2007.

Cf. Pascal (1,m,1) array: A123562 (m = -3), 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 - k, k - j]*Binomial[k, j]*(-1)^(k - j), {j, 0, n}]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 15 2017 *)

for(n=0,10, for(k=0,n, print1(sum(i=0,k, binomial(n-k, k-i) *binomial(k, i)*(-1)^(k-i)), ", "))) \\ G. C. Greubel, Oct 15 2017

T(n, k) = Sum_{i=0..k} binomial(n-k, k-i)*binomial(k, i)*(-1)^(k-i), k<=n.
T(n, k) = T(n-1, k) + T(n-1, k-1) - 2*T(n-2, k-1) (n>0). - Paul Barry, Sep 24 2004
T(n, k) = [k<=n]*Hypergeometric2F1(-k,k-n;1;-1). - Paul Barry, Jan 24 2011
E.g.f. for the n-th subdiagonal: exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} (-1)^k*binomial(n,k)* x^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 - 2*x + x^2/2) = 1 - x - 2*x^2/2! - 2*x^3/3! - x^4/4! + x^5/5! + .... - Peter Bala, Mar 05 2017

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)

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)

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

A143683 Pascal-(1,8,1) array.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 19, 19, 1, 1, 28, 118, 28, 1, 1, 37, 298, 298, 37, 1, 1, 46, 559, 1540, 559, 46, 1, 1, 55, 901, 4483, 4483, 901, 55, 1, 1, 64, 1324, 9856, 21286, 9856, 1324, 64, 1, 1, 73, 1828, 18388, 67006, 67006, 18388, 1828, 73, 1, 1, 82, 2413, 30808, 164242, 304300, 164242, 30808, 2413, 82, 1

Offset: 0

Paul Barry, Aug 28 2008

			Square array begins as:
  1,  1,    1,     1,      1,       1,        1, ... A000012;
  1, 10,   19,    28,     37,      46,       55, ... A017173;
  1, 19,  118,   298,    559,     901,     1324, ...
  1, 28,  298,  1540,   4483,    9856,    18388, ...
  1, 37,  559,  4483,  21286,   67006,   164242, ...
  1, 46,  901,  9856,  67006,  304300,  1004590, ...
  1, 55, 1324, 18388, 164242, 1004590,  4443580, ...
Antidiagonal triangle begins as:
  1;
  1,  1;
  1, 10,   1;
  1, 19,  19,    1;
  1, 28, 118,   28,    1;
  1, 37, 298,  298,   37,   1;
  1, 46, 559, 1540,  559,  46,  1;
  1, 55, 901, 4483, 4483, 901, 55, 1;

Reinhard Zumkeller, Rows n = 0..125 of table, 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), A081582 (m = 7).

Cf. A003683, A017173, A143680.

a143683 n k = a143683_tabl !! n !! k
a143683_row n = a143683_tabl !! n
a143683_tabl = map fst $ iterate
   (\(us, vs) -> (vs, zipWith (+) (map (* 8) ([0] ++ us ++ [0])) $
                      zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
-- Reinhard Zumkeller, Mar 16 2014

A143683:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A143683(n,k,8): k in [0..n], n in [0..12]]; // G. C. Greubel, May 27 2021

Table[Hypergeometric2F1[-k, k-n, 1, 9], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)

flatten([[hypergeometric([-k, k-n], [1], 9).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2021

Square array: T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 8*T(n-1, k-1) + T(n-1, k).
Number triangle: T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*9^j.
Rows are the expansions of (1+8*x)^k/(1-x)^(k+1).
Riordan array (1/(1-x), x*(1+8*x)/(1-x)).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 9). - 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)*(9*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 18*x + 81*x^2/2) = 1 + 19*x + 118*x^2/2! + 298*x^3/3! + 559*x^4/4! + 901*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n,k) = A003683(n+1). - G. C. Greubel, May 27 2021

cp's OEIS Frontend

A081589 Third row of Pascal-(1,5,1) array A081580.

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A081590 Fourth row of Pascal-(1,5,1) array A081580.

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

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A098593 A triangle of Krawtchouk coefficients.

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

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

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

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