A119900: Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.
A119900 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence from A119900
\(P_{n}(x) = \left(2\,x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 0 | 2 | 2 | |||||||||||||
| 2 | 0 | 1 | 3 | 4 | ||||||||||||
| 3 | 0 | 0 | 4 | 4 | 8 | |||||||||||
| 4 | 0 | 0 | 1 | 10 | 5 | 16 | ||||||||||
| 5 | 0 | 0 | 0 | 6 | 20 | 6 | 32 | |||||||||
| 6 | 0 | 0 | 0 | 1 | 21 | 35 | 7 | 64 | ||||||||
| 7 | 0 | 0 | 0 | 0 | 8 | 56 | 56 | 8 | 128 | |||||||
| 8 | 0 | 0 | 0 | 0 | 1 | 36 | 126 | 84 | 9 | 256 | ||||||
| 9 | 0 | 0 | 0 | 0 | 0 | 10 | 120 | 252 | 120 | 10 | 512 | |||||
| 10 | 0 | 0 | 0 | 0 | 0 | 1 | 55 | 330 | 462 | 165 | 11 | 1024 | ||||
| 11 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 220 | 792 | 792 | 220 | 12 | 2048 | |||
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 78 | 715 | 1716 | 1287 | 286 | 13 | 4096 | ||
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 14 | 364 | 2002 | 3432 | 2002 | 364 | 14 | 8192 | |
| 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 105 | 1365 | 5005 | 6435 | 3003 | 455 | 15 | 16384 |
b-files: A000079 (row sums), A034867 (reversed triangle), A109447, A119900, A202064 (reversed triangle) ?
Compute more
?
Technical details
- Node id
n-e8e64448d4c6498a· revision 0 · JSON- Definition source
oeis_catalog:A119900- Fingerprint
f6-937db0732045a112
Observed on the stored rows ?
- Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
- Palindromic rows: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): yes - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): no
Matches ?
- A202064 reversed triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
- A109447 triangle 64 terms, 15 rows · tabf · nonn · catalog b-file?
- A119900 triangle 86 terms, 12 rows · tabl · nonn · catalog b-file?
- A034867 reversed triangle 64 terms, 15 rows · tabf · nonn · catalog b-file?
Row sums ? A050732, A000079, A011782, A034008, A220051, A166444, A118655, A131577
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 0, 2; 2: 0, 1, 3; 3: 0, 0, 4, 4; 4: 0, 0, 1, 10, 5; 5: 0, 0, 0, 6, 20, 6; 6: 0, 0, 0, 1, 21, 35, 7; 7: 0, 0, 0, 0, 8, 56, 56, 8; 8: 0, 0, 0, 0, 1, 36, 126, 84, 9; 9: 0, 0, 0, 0, 0, 10, 120, 252, 120, 10;
%F FormulaThe row polynomials satisfy P(n,x) = 2*x*P(n-1,x) + (x-x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] + (x - x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] + (x - x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
A034867 reversed triangle, offset 0, from P0, low zeros dropped
%e ExampleTriangle begins: 0: 1; 1: 2; 2: 3, 1; 3: 4, 4; 4: 5, 10, 1; 5: 6, 20, 6; 6: 7, 35, 21, 1; 7: 8, 56, 56, 8; 8: 9, 84, 126, 36, 1; 9: 10, 120, 252, 120, 10;
%F FormulaRow n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = 2*x*P(n-1,x) + (x-x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] + (x - x^2)*P[n - 2, x]];
Table[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] + (x - x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
A109447 triangle, offset 1, from P0, low zeros dropped
%e ExampleTriangle begins: 1: 1; 2: 2; 3: 1, 3; 4: 4, 4; 5: 1, 10, 5; 6: 6, 20, 6; 7: 1, 21, 35, 7; 8: 8, 56, 56, 8; 9: 1, 36, 126, 84, 9; 10: 10, 120, 252, 120, 10;
%F FormulaRow n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = 2*x*P(n-1,x) + (x-x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] + (x - x^2)*P[n - 2, x]];
Table[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = 2*x*v[i - 1] + (x - x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
A119900 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 0, 2; 2: 0, 1, 3; 3: 0, 0, 4, 4; 4: 0, 0, 1, 10, 5; 5: 0, 0, 0, 6, 20, 6; 6: 0, 0, 0, 1, 21, 35, 7; 7: 0, 0, 0, 0, 8, 56, 56, 8; 8: 0, 0, 0, 0, 1, 36, 126, 84, 9; 9: 0, 0, 0, 0, 0, 10, 120, 252, 120, 10;
%F FormulaThe row polynomials satisfy P(n,x) = 2*x*P(n-1,x) + (x-x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] + (x - x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] + (x - x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
A202064 reversed triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 2, 0; 2: 3, 1, 0; 3: 4, 4, 0, 0; 4: 5, 10, 1, 0, 0; 5: 6, 20, 6, 0, 0, 0; 6: 7, 35, 21, 1, 0, 0, 0; 7: 8, 56, 56, 8, 0, 0, 0, 0; 8: 9, 84, 126, 36, 1, 0, 0, 0, 0; 9: 10, 120, 252, 120, 10, 0, 0, 0, 0, 0;
%F FormulaRow n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = 2*x*P(n-1,x) + (x-x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] + (x - x^2)*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] + (x - x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
Proof chains ? JSON
Related ?
Same fingerprint ?
- basis-map coefficients of A113214, shifted → Veronese 2 (j=1)
- A178822 → Veronese 2 (j=1)
- A003506 → Veronese 2 (j=1)
- A135838 → Veronese 2 (j=1)
- A159854 → Veronese 2 (j=1)
- A038208 → Veronese 2 (j=1)
- A344565 → Veronese 2 (j=1)
- A109447: Binomial coefficients C(n,k) with n-k odd, read by rows.
- A028326 → Veronese 2 (j=1)
- A007318 → Veronese 2 (j=1)
- A178821 → Veronese 2 (j=1)
- A034867: Triangle of odd-numbered terms in rows of Pascal's triangle.
- A178820 → Veronese 2 (j=1)
References
- [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003