A207605
Triangle of coefficients of polynomials u(n,x) jointly generated with A106195; see the Formula section.
Original entry on oeis.org
1, 2, 4, 1, 8, 4, 1, 16, 12, 5, 1, 32, 32, 18, 6, 1, 64, 80, 56, 25, 7, 1, 128, 192, 160, 88, 33, 8, 1, 256, 448, 432, 280, 129, 42, 9, 1, 512, 1024, 1120, 832, 450, 180, 52, 10, 1, 1024, 2304, 2816, 2352, 1452, 681, 242, 63, 11, 1, 2048, 5120, 6912, 6400, 4424, 2364, 985, 316, 75, 12, 1
Offset: 1
First five rows:
1
2
4 1
8 4 1
16 12 5 1
32 32 18 6 1
First four polynomials u(n,x): 1, 2, 4 + x, 8 + 4x + x^2.
(1, 1, 0, 0, 0, ...) DELTA (0, 0, 1, 0, 0, ...) begins:
1
1, 0
2, 0, 0
4, 1, 0, 0
8, 4, 1, 0, 0
16, 12, 5, 1, 0, 0
32, 32, 18, 6, 1, 0, 0. - _Philippe Deléham_, Mar 22 2012
-
CoeffList := p -> op(PolynomialTools:-CoefficientList(p,x)):
T := (n,k) -> binomial(n, k)*hypergeom([-k,n-k], [-n], x):
P := [seq(add(simplify(T(n,k)),k=0..n), n=0..11)]:
seq(CoeffList(p), p in P); # Peter Luschny, Feb 16 2018
-
(* First program *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := u[n - 1, x] + (x + 1) v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A207605 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A106195 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, 2^(n+1), If[k==n, 1, 2*T[n-1, k] + T[n-1, k-1] - T[n-2, k-1] ]]]; Join[{1}, Table[T[n, k], {n,0,10}, {k,0,n}]]//Flatten (* G. C. Greubel, Mar 15 2020 *)
-
from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + (x + 1)*v(n - 1, x)
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017
-
@CachedFunction
def T(n, k):
if (k<0 or k>n): return 0
elif k == 0: return 2^(n+1)
elif k == n: return 1
else: return 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1)
[1]+[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 15 2020
A208510
Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 7, 9, 5, 1, 1, 9, 16, 14, 6, 1, 1, 11, 25, 30, 20, 7, 1, 1, 13, 36, 55, 50, 27, 8, 1, 1, 15, 49, 91, 105, 77, 35, 9, 1, 1, 17, 64, 140, 196, 182, 112, 44, 10, 1, 1, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 1, 21, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1
Offset: 1
First five rows:
1
1...1
1...3...1
1...5...4...1
1...7...9...5...1
First five polynomials u(n,x):
1
1 + x
1 + 3x + x^2
1 + 5x + 4x^2 + x^3
1 + 7x + 9x^2 + 5x^3 + x^4
-
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A208510 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A029653 *)
-
from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017
A029653
Numbers in (2,1)-Pascal triangle (by row).
Original entry on oeis.org
1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 7, 9, 5, 1, 2, 9, 16, 14, 6, 1, 2, 11, 25, 30, 20, 7, 1, 2, 13, 36, 55, 50, 27, 8, 1, 2, 15, 49, 91, 105, 77, 35, 9, 1, 2, 17, 64, 140, 196, 182, 112, 44, 10, 1, 2, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 2, 21, 100, 285
Offset: 0
The triangle T(n,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 2 1
2: 2 3 1
3: 2 5 4 1
4: 2 7 9 5 1
5: 2 9 16 14 6 1
6: 2 11 25 30 20 7 1
7: 2 13 36 55 50 27 8 1
8: 2 15 49 91 105 77 35 9 1
9: 2 17 64 140 196 182 112 44 10 1
10: 2 19 81 204 336 378 294 156 54 11 1
... Reformatted. - _Wolfdieter Lang_, Jan 09 2015
With the array M(k) as defined in the Formula section, the infinite product M(0)*M(1)*M(2)*... begins
/1 \/1 \/1 \ /1 \
|2 1 ||0 1 ||0 1 | |2 1 |
|2 1 1 ||0 2 1 ||0 0 1 |... = |2 3 1 |
|2 1 1 1 ||0 2 1 1 ||0 0 2 1 | |2 5 4 1 |
|2 1 1 1 1||0 2 1 1 1 ||0 0 2 1 1| |2 7 9 5 1|
|... ||... ||... | |... |
- _Peter Bala_, Dec 27 2014
- Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
- Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
- Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
- Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
- Hacene Belbachir and Athmane Benmezai, Expansion of Fibonacci and Lucas Polynomials: An Answer to Prodinger's Question, Journal of Integer Sequences, Vol. 15 (2012), #12.7.6.
- Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 39.
- H. Hosoya, Pascal's triangle, non-adjacent numbers and D-dimensional atomic orbitals, J. Math. Chemistry, vol. 23, 1998, 169-178.
- M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
- M. Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5
- Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 8.
- Mark C. Wilson, Asymptotics for generalized Riordan arrays. International Conference on Analysis of Algorithms DMTCS proc. AD. Vol. 323. 2005.
-
a029653 n k = a029653_tabl !! n !! k
a029653_row n = a029653_tabl !! n
a029653_tabl = [1] : iterate
(\xs -> zipWith (+) ([0] ++ xs) (xs ++ [0])) [2, 1]
-- Reinhard Zumkeller, Dec 16 2013
-
A029653 := proc(n,k)
if n = 0 then
1;
else
binomial(n-1, k)+binomial(n, k)
fi
end proc: # R. J. Mathar, Jun 30 2013
-
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A208510 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A029653 *)
(* Clark Kimberling, Feb 28 2012 *)
-
from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1
def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017
-
from math import comb, isqrt
def A029653(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),a:=n-comb(r+1,2))*((r<<1)-a)//r if n else 1 # Chai Wah Wu, Nov 12 2024
A078812
Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
Original entry on oeis.org
1, 2, 1, 3, 4, 1, 4, 10, 6, 1, 5, 20, 21, 8, 1, 6, 35, 56, 36, 10, 1, 7, 56, 126, 120, 55, 12, 1, 8, 84, 252, 330, 220, 78, 14, 1, 9, 120, 462, 792, 715, 364, 105, 16, 1, 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1, 11, 220, 1287, 3432, 5005, 4368, 2380, 816, 171, 20, 1
Offset: 0
Triangle begins, 1 <= k <= n:
1
2 1
3 4 1
4 10 6 1
5 20 21 8 1
6 35 56 36 10 1
7 56 126 120 55 12 1
8 84 252 330 220 78 14 1
From _Peter Bala_, Feb 11 2025: (Start)
The array factorizes as an infinite product of lower triangular arrays:
/ 1 \ / 1 \ / 1 \ / 1 \
| 2 1 | | 2 1 | | 0 1 | | 0 1 |
| 3 4 1 | = | 3 2 1 | | 0 2 1 | | 0 0 1 | ...
| 4 10 6 1 | | 4 3 2 1 | | 0 3 2 1 | | 0 0 2 1 |
| 5 20 21 8 1| | 5 4 3 2 1| | 0 4 3 2 1 | | 0 0 3 2 1 |
|... | |... | |... | |... |
Cf. A092276. (End)
- T. D. Noe, Rows n = 0..50 of triangle, flattened
- J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - From _N. J. A. Sloane_, May 10 2012
- Peter Bala, Factorisations of some Riordan arrays as infinite products
- Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
- Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
- Paul Barry, Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers, arXiv:2011.10827 [math.CO], 2020.
- P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
- Nour-Eddine Fahssi, On the combinatorics of exclusion in Haldane fractional statistics, arXiv:1808.00045 [cond-mat.stat-mech], 2018.
- Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
- A. Laradji, and A. Umar, Combinatorial results for semigroups of order-preserving full transformations, Semigroup Forum 72 (2006), 51-62.
- Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart. 43, no. 4, (2005) 359-370.
This triangle is formed from odd-numbered rows of triangle
A011973 read in reverse order.
-
Flat(List([0..12], n-> List([0..n], k-> Binomial(n+k+1, 2*k+1) ))); # G. C. Greubel, Aug 01 2019
-
a078812 n k = a078812_tabl !! n !! k
a078812_row n = a078812_tabl !! n
a078812_tabl = [1] : [2, 1] : f [1] [2, 1] where
f us vs = ws : f vs ws where
ws = zipWith (-) (zipWith (+) ([0] ++ vs) (map (* 2) vs ++ [0]))
(us ++ [0, 0])
-- Reinhard Zumkeller, Dec 16 2013
-
/* As triangle */ [[Binomial(n+k-1, 2*k-1): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jun 01 2018
-
for n from 1 to 11 do seq(binomial(n+k-1,2*k-1),k=1..n) od; # yields sequence in triangular form; Emeric Deutsch, Apr 09 2005
# Uses function PMatrix from A357368. Adds a row and column above and to the left.
PMatrix(10, n -> n); # Peter Luschny, Oct 07 2022
-
(* First program *)
u[1, x_]:= 1; v[1, x_]:= 1; z = 13;
u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
v[n_, x_]:= u[n-1, x] + (x+1)*v[n-1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A085478 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A078812 *) (* Clark Kimberling, Feb 25 2012 *)
(* Second program *)
Table[Binomial[n+k+1, 2*k+1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 01 2019 *)
-
T(n,m):=sum(binomial(2*k,n-m)*binomial(m+k,k)*(-1)^(n-m+k)*binomial(n+1,m+k+1),k,0,n-m); /* Vladimir Kruchinin, Apr 13 2016 */
-
{T(n, k) = if( n<0, 0, binomial(n+k-1, 2*k-1))};
-
{T(n, k) = polcoeff( polcoeff( x*y / (1 - (2 + y) * x + x^2) + x * O(x^n), n), k)};
-
@cached_function
def T(k,n):
if k==n: return 1
if k==0: return 0
return sum(i*T(k-1,n-i) for i in (1..n-k+1))
A078812 = lambda n,k: T(k,n)
[[A078812(n,k) for k in (1..n)] for n in (1..8)] # Peter Luschny, Mar 12 2016
-
[[binomial(n+k+1, 2*k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019
A208341
Triangle read by rows, T(n,k) = hypergeometric_2F1([n-k+1, -k], [1], -1) for n>=0 and k>=0.
Original entry on oeis.org
1, 1, 2, 1, 3, 4, 1, 4, 8, 8, 1, 5, 13, 20, 16, 1, 6, 19, 38, 48, 32, 1, 7, 26, 63, 104, 112, 64, 1, 8, 34, 96, 192, 272, 256, 128, 1, 9, 43, 138, 321, 552, 688, 576, 256, 1, 10, 53, 190, 501, 1002, 1520, 1696, 1280, 512, 1, 11, 64, 253, 743, 1683, 2972, 4048
Offset: 0
First five rows:
1;
1, 2;
1, 3, 4;
1, 4, 8, 8;
1, 5, 13, 20, 16;
First five polynomials v(n,x):
1
1 + 2x
1 + 3x + 4x^2
1 + 4x + 8x^2 + 8x^3
1 + 5x + 13x^2 + 20x^3 + 16x^4
(1, 0, -1/2, 1/2, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins:
1;
1, 0;
1, 2, 0;
1, 3, 4, 0;
1, 4, 8, 8, 0;
1, 5, 13, 20, 16, 0;
1, 6, 19, 38, 48, 32, 0;
Triangle in A049600 begins:
0;
0, 1;
0, 1, 2;
0, 1, 3, 4;
0, 1, 4, 8, 8;
0, 1, 5, 13, 20, 16;
0, 1, 6, 19, 38, 48, 32;
... - _Philippe Deléham_, Mar 23 2012
-
a208341 n k = a208341_tabl !! (n-1) !! (k-1)
a208341_row n = a208341_tabl !! (n-1)
a208341_tabl = map reverse a106195_tabl
-- Reinhard Zumkeller, Dec 16 2013
-
T := (n,k) -> hypergeom([n-k+1, -k],[1],-1):
seq(lprint(seq(simplify(T(n,k)),k=0..n)),n=0..7); # Peter Luschny, May 20 2015
-
u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2*x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A160232 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208341 *)
-
T(n,k) = sum(i = 0, k, 2^(k-i)*binomial(n-k,i)*binomial(k,i));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print();); \\ Michel Marcus, Aug 14 2015
A124448
Riordan array (sqrt(1+4x^2)-2x, (1+2x-sqrt(1+4x^2))/2).
Original entry on oeis.org
1, -2, 1, 2, -3, 1, 0, 4, -4, 1, -2, -1, 7, -5, 1, 0, -4, -4, 11, -6, 1, 4, 2, -6, -10, 16, -7, 1, 0, 8, 8, -6, -20, 22, -8, 1, -10, -5, 11, 19, -1, -35, 29, -9, 1, 0, -20, -20, 7, 34, 13, -56, 37, -10, 1, 28, 14, -26, -46, -12, 49, 41, -84
Offset: 0
Triangle begins
1;
-2, 1;
2, -3, 1;
0, 4, -4, 1;
-2, -1, 7, -5, 1;
0, -4, -4, 11, -6, 1;
4, 2, -6, -10, 16, -7, 1;
0, 8, 8, -6, -20, 22, -8, 1;
-
N=12;
T(n, k)=sum(i=0, n-k, binomial(k, i)*binomial(n-k, i)*2^(n-k-i));
M=matrix(N, N);
for(n=1, N, for(k=1, n, M[n, k]=T(n-1, k-1))); /* A106195 */
A=M^-1; /* A124448 */
/* for (n=1, N, for(k=1, n, print1(M[n, k], ", "))); */ /* A106195 */
for (n=1, N, for(k=1, n, print1(A[n, k], ", "))); /* A124448 */
/* Joerg Arndt, May 14 2011 */
Showing 1-6 of 6 results.
Comments