A209145
Triangle of coefficients of polynomials u(n,x) jointly generated with A122075; see the Formula section.
Original entry on oeis.org
1, 2, 1, 4, 4, 1, 7, 10, 5, 1, 12, 22, 16, 6, 1, 20, 45, 43, 23, 7, 1, 33, 88, 104, 72, 31, 8, 1, 54, 167, 235, 199, 110, 40, 9, 1, 88, 310, 506, 506, 340, 158, 50, 10, 1, 143, 566, 1051, 1211, 956, 538, 217, 61, 11, 1, 232, 1020, 2123, 2768, 2507, 1652, 805, 288, 73, 12, 1
Offset: 1
First five rows:
1
2 1
4 4 1
7 10 5 1
12 22 16 6 1
First three polynomials u(n,x): 1, 2 + x, 4 + 4*x + x^2.
-
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*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[%] (* A209145 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A122075 *)
A236076
A skewed version of triangular array A122075.
Original entry on oeis.org
1, 0, 2, 0, 1, 3, 0, 0, 3, 5, 0, 0, 1, 7, 8, 0, 0, 0, 4, 15, 13, 0, 0, 0, 1, 12, 30, 21, 0, 0, 0, 0, 5, 31, 58, 34, 0, 0, 0, 0, 1, 18, 73, 109, 55, 0, 0, 0, 0, 0, 6, 54, 162, 201, 89, 0, 0, 0, 0, 0, 1, 25, 145, 344, 365, 144, 0, 0, 0, 0, 0, 0, 7, 85, 361
Offset: 0
Triangle begins:
1;
0, 2;
0, 1, 3;
0, 0, 3, 5;
0, 0, 1, 7, 8;
0, 0, 0, 4, 15, 13;
0, 0, 0, 1, 12, 30, 21;
0, 0, 0, 0, 5, 31, 58, 34;
-
a236076 n k = a236076_tabl !! n !! k
a236076_row n = a236076_tabl !! n
a236076_tabl = [1] : [0, 2] : f [1] [0, 2] where
f us vs = ws : f vs ws where
ws = [0] ++ zipWith (+) (zipWith (+) ([0] ++ us) (us ++ [0])) vs
-- Reinhard Zumkeller, Jan 19 2014
-
T[n_, k_]:= If[k<0 || k>n, 0, If[n==0 && k==0, 1, If[k==0, 0, If[n==1 && k==1, 2, T[n-1, k-1] + T[n-2, k-1] + T[n-2, k-2]]]]]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, May 21 2019 *)
-
{T(n,k) = if(k<0 || k>n, 0, if(n==0 && k==0, 1, if(k==0, 0, if(n==1 && k==1, 2, T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2) ))))}; \\ G. C. Greubel, May 21 2019
-
def T(n, k):
if (k<0 or k>n): return 0
elif (n==0 and k==0): return 1
elif (k==0): return 0
elif (n==1 and k==1): return 2
else: return T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 21 2019
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
A037027
Skew Fibonacci-Pascal triangle read by rows.
Original entry on oeis.org
1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 5, 10, 9, 4, 1, 8, 20, 22, 14, 5, 1, 13, 38, 51, 40, 20, 6, 1, 21, 71, 111, 105, 65, 27, 7, 1, 34, 130, 233, 256, 190, 98, 35, 8, 1, 55, 235, 474, 594, 511, 315, 140, 44, 9, 1, 89, 420, 942, 1324, 1295, 924, 490, 192, 54, 10, 1, 144, 744, 1836
Offset: 0
Ratio of row polynomials R(3)/R(2) = (3 + 5*x + 3*x^2 + x^3)/(2 + 2*x + x^2) = [1+x; 1+x, 1+x].
Triangle begins:
1;
1, 1;
2, 2, 1;
3, 5, 3, 1;
5, 10, 9, 4, 1;
8, 20, 22, 14, 5, 1;
13, 38, 51, 40, 20, 6, 1;
21, 71, 111, 105, 65, 27, 7, 1;
34, 130, 233, 256, 190, 98, 35, 8, 1;
55, 235, 474, 594, 511, 315, 140, 44, 9, 1;
89, 420, 942, 1324, 1295, 924, 490, 192, 54, 10, 1;
- Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
- Harlan J. Brothers, Pascal's Prism: Supplementary Material
- Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
- T. Mansour, Generalization of some identities involving the Fibonacci numbers, arXiv:math/0301157 [math.CO], 2003.
- P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
- Yidong Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, Nov. 2005, pp. 359-370.
- Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials.
-
a037027 n k = a037027_tabl !! n !! k
a037027_row n = a037027_tabl !! n
a037027_tabl = [1] : [1,1] : f [1] [1,1] where
f xs ys = ys' : f ys ys' where
ys' = zipWith3 (\u v w -> u + v + w) (ys ++ [0]) (xs ++ [0,0]) ([0] ++ ys)
-- Reinhard Zumkeller, Jul 07 2012
-
T := (n,k) -> `if`(n=0,1,binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -4)): seq(seq(simplify(T(n,k)), k=0..n), n=0..10); # Peter Luschny, Apr 25 2016
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> combinat:-fibonacci(n)); # Peter Luschny, Oct 07 2022
-
Mv[x, -1] = 0; Mv[x, 0] = 1; Mv[x, 1] = 1 + x; Mv[x_, n_] := Mv[x, n] = ExpandAll[(x + 1)*Mv[x, n - 1] + Mv[x, n - 2]]; Table[ CoefficientList[ Mv[x, n], x], {n, 0, 10}] // Flatten (* Roger L. Bagula, Apr 09 2008 *)
Abs[Flatten[Table[CoefficientList[CharacteristicPolynomial[Array[KroneckerDelta[#1,#2]+KroneckerDelta[#1,#2+1]*I+KroneckerDelta[#1,#2-1]*I&,{n,n}],x],x],{n,1,20}]]] (* John M. Campbell, Aug 23 2011 *)
T[n_, k_] := Binomial[n, k] Hypergeometric2F1[(k-n)/2, (k-n+1)/2, -n, -4];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 16 2019, after Peter Luschny *)
-
{T(n, k) = if( k<0 || k>n, 0, if( n==0 && k==0, 1, T(n-1, k) + T(n-1, k-1) + T(n-2, k)))}; /* Michael Somos, Sep 29 2003 */
-
T(n,k)=if(nPaul D. Hanna, Feb 27 2004
A055830
Triangle T read by rows: diagonal differences of triangle A037027.
Original entry on oeis.org
1, 1, 0, 2, 1, 0, 3, 3, 1, 0, 5, 7, 4, 1, 0, 8, 15, 12, 5, 1, 0, 13, 30, 31, 18, 6, 1, 0, 21, 58, 73, 54, 25, 7, 1, 0, 34, 109, 162, 145, 85, 33, 8, 1, 0, 55, 201, 344, 361, 255, 125, 42, 9, 1, 0, 89, 365, 707, 850, 701, 413, 175, 52, 10, 1, 0, 144, 655, 1416, 1918, 1806, 1239, 630, 236, 63, 11, 1, 0
Offset: 0
Triangle begins:
1
1, 0
2, 1, 0
3, 3, 1, 0
5, 7, 4, 1, 0
8, 15, 12, 5, 1, 0
13, 30, 31, 18, 6, 1, 0
21, 58, 73, 54, 25, 7, 1, 0
34, 109, 162, 145, 85, 33, 8, 1, 0
55, 201, 344, 361, 255, 125, 42, 9, 1, 0
...
Row sums:
A001333 (numerators of continued fraction convergents to sqrt(2)).
-
function T(n,k)
if k lt 0 or k gt n then return 0;
elif k eq 0 then return Fibonacci(n+1);
elif n eq 1 and k eq 1 then return 0;
else return T(n-1,k-1) + T(n-1,k) + T(n-2,k);
end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 21 2020
-
with(combinat);
T:= proc(n, k) option remember;
if k<0 or k>n then 0
elif k=0 then fibonacci(n+1)
elif n=1 and k=1 then 0
else T(n-1, k-1) + T(n-1, k) + T(n-2, k)
fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Jan 21 2020
-
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, Fibonacci[n+1], If[n==1 && k==1, 0, T[n-1, k-1] + T[n-1, k] + T[n-2, k]]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 19 2017 *)
-
T(n,k) = if(k<0 || k>n, 0, if(k==0, fibonacci(n+1), if(n==1 && k==1, 0, T(n-1, k-1) + T(n-1, k) + T(n-2, k) )));
for(n=0,12, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jan 21 2020
-
@CachedFunction
def T(n, k):
if (k<0 or k>n): return 0
elif (k==0): return fibonacci(n+1)
elif (n==1 and k==1): return 0
else: return T(n-1, k-1) + T(n-1, k) + T(n-2, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 21 2020
A374439
Triangle read by rows: the coefficients of the Lucas-Fibonacci polynomials. T(n, k) = T(n - 1, k) + T(n - 2, k - 2) with initial values T(n, k) = k + 1 for k < 2.
Original entry on oeis.org
1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 3, 4, 1, 1, 2, 4, 6, 3, 2, 1, 2, 5, 8, 6, 6, 1, 1, 2, 6, 10, 10, 12, 4, 2, 1, 2, 7, 12, 15, 20, 10, 8, 1, 1, 2, 8, 14, 21, 30, 20, 20, 5, 2, 1, 2, 9, 16, 28, 42, 35, 40, 15, 10, 1, 1, 2, 10, 18, 36, 56, 56, 70, 35, 30, 6, 2
Offset: 0
Triangle starts:
[ 0] [1]
[ 1] [1, 2]
[ 2] [1, 2, 1]
[ 3] [1, 2, 2, 2]
[ 4] [1, 2, 3, 4, 1]
[ 5] [1, 2, 4, 6, 3, 2]
[ 6] [1, 2, 5, 8, 6, 6, 1]
[ 7] [1, 2, 6, 10, 10, 12, 4, 2]
[ 8] [1, 2, 7, 12, 15, 20, 10, 8, 1]
[ 9] [1, 2, 8, 14, 21, 30, 20, 20, 5, 2]
[10] [1, 2, 9, 16, 28, 42, 35, 40, 15, 10, 1]
.
Table of interpolated sequences:
| n | A039834 & A000045 | A000032 | A000129 | A048654 |
| n | -P(n,-1) | P(n,1) |2^n*P(n,-1/2)|2^n*P(n,1/2)|
| | Fibonacci | Lucas | Pell | Pell* |
| 0 | -1 | 1 | 1 | 1 |
| 1 | 1 | 3 | 0 | 4 |
| 2 | 0 | 4 | 1 | 9 |
| 3 | 1 | 7 | 2 | 22 |
| 4 | 1 | 11 | 5 | 53 |
| 5 | 2 | 18 | 12 | 128 |
| 6 | 3 | 29 | 29 | 309 |
| 7 | 5 | 47 | 70 | 746 |
| 8 | 8 | 76 | 169 | 1801 |
| 9 | 13 | 123 | 408 | 4348 |
Adding and subtracting the values in a row of the table (plus halving the values obtained in this way):
A022087,
A055389,
A118658,
A052542,
A163271,
A371596,
A324969,
A212804,
A077985,
A069306,
A215928.
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function T(n,k) // T = A374439
if k lt 0 or k gt n then return 0;
elif k le 1 then return k+1;
else return T(n-1,k) + T(n-2,k-2);
end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 23 2025
-
A374439 := (n, k) -> ifelse(k::odd, 2, 1)*binomial(n - irem(k, 2) - iquo(k, 2), iquo(k, 2)):
# Alternative, using the function qStirling2 from A333143:
T := (n, k) -> 2^irem(k, 2)*qStirling2(n, k, -1):
seq(seq(T(n, k), k = 0..n), n = 0..10);
-
A374439[n_, k_] := (# + 1)*Binomial[n - (k + #)/2, (k - #)/2] & [Mod[k, 2]];
Table[A374439[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Paolo Xausa, Jul 24 2024 *)
-
from functools import cache
@cache
def T(n: int, k: int) -> int:
if k > n: return 0
if k < 2: return k + 1
return T(n - 1, k) + T(n - 2, k - 2)
-
from math import comb as binomial
def T(n: int, k: int) -> int:
o = k & 1
return binomial(n - o - (k - o) // 2, (k - o) // 2) << o
-
def P(n, x):
if n < 0: return P(n, x)
return sum(T(n, k)*x**k for k in range(n + 1))
def sgn(x: int) -> int: return (x > 0) - (x < 0)
# Table of interpolated sequences
print("| n | A039834 & A000045 | A000032 | A000129 | A048654 |")
print("| n | -P(n,-1) | P(n,1) |2^n*P(n,-1/2)|2^n*P(n,1/2)|")
print("| | Fibonacci | Lucas | Pell | Pell* |")
f = "| {0:2d} | {1:9d} | {2:4d} | {3:5d} | {4:4d} |"
for n in range(10): print(f.format(n, -P(n, -1), P(n, 1), int(2**n*P(n, -1/2)), int(2**n*P(n, 1/2))))
-
from sage.combinat.q_analogues import q_stirling_number2
def A374439(n,k): return (-1)^((k+1)//2)*2^(k%2)*q_stirling_number2(n+1, k+1, -1)
print(flatten([[A374439(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 23 2025
A209599
Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Original entry on oeis.org
1, 2, 0, 3, 1, 0, 5, 3, 0, 0, 8, 7, 1, 0, 0, 13, 15, 4, 0, 0, 0, 21, 30, 12, 1, 0, 0, 0, 34, 58, 31, 5, 0, 0, 0, 0, 55, 109, 73, 18, 1, 0, 0, 0, 0, 89, 201, 162, 54, 6, 0, 0, 0, 0, 0, 144, 365, 344, 145, 25, 1, 0, 0, 0, 0, 0
Offset: 0
Triangle begins :
1
2, 0
3, 1, 0
5, 3, 0, 0
8, 7, 1, 0, 0
13, 15, 4, 0, 0, 0
21, 30, 12, 1, 0, 0, 0
34, 58, 31, 5, 0, 0, 0, 0
55, 109, 73, 18, 1, 0, 0, 0, 0
89, 201, 162, 54, 6, 0, 0, 0, 0, 0
144, 365, 344, 145, 25, 1, 0, 0, 0, 0, 0
...
-
T[0, 0] := 1; T[1, 0] := 2; T[1, 1] := 0; T[n_, k_] := T[n, k] = If[n<0, 0, If[k > n, 0, T[n - 1, k] + T[n - 2, k] + T[n - 2, k - 1]]]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 19 2017 *)
Showing 1-7 of 7 results.
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