A131222: Exponential Riordan array [1, log((1-x)/(1-2x))].
A131222 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence from A131222
\(P_{n}(x) = \left(-3 + x + 3\,n\right)\,P_{n-1}(x) + \left(-4 + 6\,n - 2\,n^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 0 | 1 | 1 | ||||||
| 2 | 0 | 3 | 1 | 4 | |||||
| 3 | 0 | 14 | 9 | 1 | 24 | ||||
| 4 | 0 | 90 | 83 | 18 | 1 | 192 | |||
| 5 | 0 | 744 | 870 | 275 | 30 | 1 | 1920 | ||
| 6 | 0 | 7560 | 10474 | 4275 | 685 | 45 | 1 | 23040 | |
| 7 | 0 | 91440 | 143892 | 70924 | 14805 | 1435 | 63 | 1 | 322560 |
All 15 rows b-files: A002866 (row sums), A079638, A131222 ?
Compute more
?
Technical details
- Node id
n-e41c30a965e7cb9c· revision 0 · JSON- Definition source
oeis_catalog:A131222- Fingerprint
f6-bd19e0a95c26bd62
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 ?
- A131222 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?
- A079638 triangle 43 terms, 8 rows · tabl · nonn · catalog b-file?
Row sums ? A002866, A052656, A121892, A052594, A032107
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 0, 1; 2: 0, 3, 1; 3: 0, 14, 9, 1; 4: 0, 90, 83, 18, 1; 5: 0, 744, 870, 275, 30, 1; 6: 0, 7560, 10474, 4275, 685, 45, 1; 7: 0, 91440, 143892, 70924, 14805, 1435, 63, 1; 8: 0, 1285200, 2233356, 1274196, 324289, 41160, 2674, 84, 1; 9: 0, 20603520, 38769840, 24870572, 7398972, 1151409, 98280, 4578, 108, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (3*n-3+x)*P(n-1,x) - (2*n^2-6*n+4)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[(3*n - 3 + x)*P[n - 1, x] - (2*n^2 - 6*n + 4)*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] = x; for(n = 2, N, my(i = n + 1); v[i] = (3*n - 3 + x)*v[i - 1] - (2*n^2 - 6*n + 4)*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
A079638 triangle, offset 1, from P1, low zeros dropped
%e ExampleTriangle begins: 1: 1; 2: 3, 1; 3: 14, 9, 1; 4: 90, 83, 18, 1; 5: 744, 870, 275, 30, 1; 6: 7560, 10474, 4275, 685, 45, 1; 7: 91440, 143892, 70924, 14805, 1435, 63, 1; 8: 1285200, 2233356, 1274196, 324289, 41160, 2674, 84, 1; 9: 20603520, 38769840, 24870572, 7398972, 1151409, 98280, 4578, 108, 1; 10: 371226240, 745785936, 526791060, 177945200, 32559975, 3419913, 209790, 7350, 135, 1;
%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) = (3*n-3+x)*P(n-1,x) - (2*n^2-6*n+4)*P(n-2,x) for n >= 3, with P(1,x) = x, P(2,x) = 3*x + x^2.
%t MathematicaP[1, x_] := x;
P[2, x_] := 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[(3*n - 3 + x)*P[n - 1, x] - (2*n^2 - 6*n + 4)*P[n - 2, x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = x; v[2] = 3*x + x^2; for(n = 3, N, my(i = n); v[i] = (3*n - 3 + x)*v[i - 1] - (2*n^2 - 6*n + 4)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
A131222 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 0, 1; 2: 0, 3, 1; 3: 0, 14, 9, 1; 4: 0, 90, 83, 18, 1; 5: 0, 744, 870, 275, 30, 1; 6: 0, 7560, 10474, 4275, 685, 45, 1; 7: 0, 91440, 143892, 70924, 14805, 1435, 63, 1; 8: 0, 1285200, 2233356, 1274196, 324289, 41160, 2674, 84, 1; 9: 0, 20603520, 38769840, 24870572, 7398972, 1151409, 98280, 4578, 108, 1;
%F FormulaThe row polynomials satisfy P(n,x) = (3*n-3+x)*P(n-1,x) - (2*n^2-6*n+4)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[(3*n - 3 + x)*P[n - 1, x] - (2*n^2 - 6*n + 4)*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] = x; for(n = 2, N, my(i = n + 1); v[i] = (3*n - 3 + x)*v[i - 1] - (2*n^2 - 6*n + 4)*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
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
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
Proof chains ? JSON
Used in 12
- basis_map_A013611 of Exponential Riordan array [1, log((1-x)/(1-2x))]. Image under a basis map u^k -> P_k (A013611)
- basis_map_A013610 of Exponential Riordan array [1, log((1-x)/(1-2x))]. Image under a basis map u^k -> P_k (A013610)
- basis_map_A013609 of Exponential Riordan array [1, log((1-x)/(1-2x))]. Image under a basis map u^k -> P_k (A013609)
- shift_x_plus_one of Exponential Riordan array [1, log((1-x)/(1-2x))]. Image under a basis map u^k -> P_k (A007318)
- basis_map_A008277 of Exponential Riordan array [1, log((1-x)/(1-2x))]. Image under a basis map u^k -> P_k (A008277)
- theta_plus_one of Exponential Riordan array [1, log((1-x)/(1-2x))]. Image under a real-rootedness preserver (euler polynomial, shift 0)
- derivative of Exponential Riordan array [1, log((1-x)/(1-2x))]. Image under a real-rootedness preserver (derivative polynomial, shift 0)
- reversal of Exponential Riordan array [1, log((1-x)/(1-2x))]. Image under a real-rootedness preserver (reversal, shift 0)
4 more
- shift_x_plus_one of Exponential Riordan array [1, log((1-x)/(1-2x))]. Image under a real-rootedness preserver (affine, shift 0)
- mobius_x_over_x_plus_one of Exponential Riordan array [1, log((1-x)/(1-2x))]. Image under a real-rootedness preserver (mobius, shift 0)
- theta_plus_one_squared of Exponential Riordan array [1, log((1-x)/(1-2x))]. Image under a real-rootedness preserver (euler polynomial, shift 0)
- one_plus_derivative of Exponential Riordan array [1, log((1-x)/(1-2x))]. Image under a real-rootedness preserver (derivative polynomial, shift 0)
Related ?
Same fingerprint ?
- A079638 → reverse
- A079638: Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.
- A131222 → reverse
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