A330381: Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= n.
A330381 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)
Recurrence from A330381
\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-1}(x) + \left(-1 - x - x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 1 | 2 | 3 | |||||||||||||
| 2 | 1 | 5 | 3 | 9 | ||||||||||||
| 3 | 1 | 9 | 13 | 4 | 27 | |||||||||||
| 4 | 1 | 14 | 35 | 26 | 5 | 81 | ||||||||||
| 5 | 1 | 20 | 75 | 96 | 45 | 6 | 243 | |||||||||
| 6 | 1 | 27 | 140 | 267 | 216 | 71 | 7 | 729 | ||||||||
| 7 | 1 | 35 | 238 | 623 | 750 | 427 | 105 | 8 | 2187 | |||||||
| 8 | 1 | 44 | 378 | 1288 | 2123 | 1800 | 770 | 148 | 9 | 6561 | ||||||
| 9 | 1 | 54 | 570 | 2436 | 5211 | 6046 | 3858 | 1296 | 201 | 10 | 19683 | |||||
| 10 | 1 | 65 | 825 | 4302 | 11505 | 17303 | 15115 | 7590 | 2067 | 265 | 11 | 59049 | ||||
| 11 | 1 | 77 | 1155 | 7194 | 23397 | 43923 | 49721 | 34210 | 13959 | 3157 | 341 | 12 | 177147 | |||
| 12 | 1 | 90 | 1573 | 11506 | 44550 | 101530 | 143365 | 127854 | 71566 | 24310 | 4653 | 430 | 13 | 531441 | ||
| 13 | 1 | 104 | 2093 | 17732 | 80366 | 217646 | 372749 | 414584 | 300950 | 140426 | 40469 | 6656 | 533 | 14 | 1594323 | |
| 14 | 1 | 119 | 2730 | 26481 | 138567 | 438438 | 891345 | 1201917 | 1088283 | 659022 | 261261 | 64857 | 9282 | 651 | 15 | 4782969 |
b-files: A000244 (row sums), A104029 (reversed triangle), A330381 ?
Compute more
?
Other recurrences
Reversed rows · proved equal for all n (proof step)
\(P_{n}(x) = \left(-1 - x - x^{2}\right)\,P_{n-2}(x) + \left(2 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
Technical details
- Node id
n-b1a7dcc8610d7e32· revision 1 · JSON- Definition source
oeis_catalog:A330381derived:reversal(equal bye-637632b6c2fd4634)- Fingerprint
f6-2f76d9ab212d3ec7
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 ?
- A104029 reversed triangle 62 terms, 10 rows · tabl · nonn · catalog b-file?
- A330381 triangle 61 terms, 10 rows · tabl · nonn · catalog b-file?
Row sums ? A140429, A354269, A133494, A155734, A027313, A000244, A376478
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 2; 2: 1, 5, 3; 3: 1, 9, 13, 4; 4: 1, 14, 35, 26, 5; 5: 1, 20, 75, 96, 45, 6; 6: 1, 27, 140, 267, 216, 71, 7; 7: 1, 35, 238, 623, 750, 427, 105, 8; 8: 1, 44, 378, 1288, 2123, 1800, 770, 148, 9; 9: 1, 54, 570, 2436, 5211, 6046, 3858, 1296, 201, 10;
%F FormulaThe row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1+x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + 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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (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
A104029 reversed triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 2, 1; 2: 3, 5, 1; 3: 4, 13, 9, 1; 4: 5, 26, 35, 14, 1; 5: 6, 45, 96, 75, 20, 1; 6: 7, 71, 216, 267, 140, 27, 1; 7: 8, 105, 427, 750, 623, 238, 35, 1; 8: 9, 148, 770, 1800, 2123, 1288, 378, 44, 1; 9: 10, 201, 1296, 3858, 6046, 5211, 2436, 570, 54, 1;
%F FormulaRow n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*x)*P(n-1,x) - (1+x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + 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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (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
A330381 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 1, 2; 2: 1, 5, 3; 3: 1, 9, 13, 4; 4: 1, 14, 35, 26, 5; 5: 1, 20, 75, 96, 45, 6; 6: 1, 27, 140, 267, 216, 71, 7; 7: 1, 35, 238, 623, 750, 427, 105, 8; 8: 1, 44, 378, 1288, 2123, 1800, 770, 148, 9; 9: 1, 54, 570, 2436, 5211, 6046, 3858, 1296, 201, 10;
%F FormulaThe row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1+x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + 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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (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
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
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Liu–Wang theorem for three-term recurrences (above)
Used in 12
- basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary str… Image under a basis map u^k -> P_k (A013610)
- basis_map_A008277 of Triangle read by rows: T(n,k) is the number of ternary str… Image under a basis map u^k -> P_k (A008277)
- basis_map_A013609 of Triangle read by rows: T(n,k) is the number of ternary str… Image under a basis map u^k -> P_k (A013609)
- shift_x_plus_one of Triangle read by rows: T(n,k) is the number of ternary stri… Image under a basis map u^k -> P_k (A007318)
- basis_map_A013611 of Triangle read by rows: T(n,k) is the number of ternary str… Image under a basis map u^k -> P_k (A013611)
- shift_x_plus_one of Triangle read by rows: T(n,k) is the number of ternary stri… Image under a real-rootedness preserver (affine, shift 0)
- theta_plus_one_squared of Triangle read by rows: T(n,k) is the number of ternar… Image under a real-rootedness preserver (euler polynomial, shift 0)
- derivative of Triangle read by rows: T(n,k) is the number of ternary strings of… Image under a real-rootedness preserver (derivative polynomial, shift 0)
4 more
- Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907). Image under a real-rootedness preserver (reversal, shift 0)
- one_plus_derivative of Triangle read by rows: T(n,k) is the number of ternary s… Image under a real-rootedness preserver (derivative polynomial, shift 0)
- mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of tern… Image under a real-rootedness preserver (mobius, shift 0)
- theta_plus_one of Triangle read by rows: T(n,k) is the number of ternary string… Image under a real-rootedness preserver (euler polynomial, shift 0)
Related ?
Same fingerprint ?
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