A089503: Triangle of numbers used for basis change between certain falling factorials.
A089503 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernel
Recurrence from A089503
\(P_{n}(x) = \left(1 - 4\,x + 2\,x^{2} + 4\,n x - 3\,n x^{2} + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-x^{2} + x^{3} - n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 4 | 5 | ||||||
| 3 | 1 | 12 | 30 | 43 | |||||
| 4 | 1 | 24 | 168 | 336 | 529 | ||||
| 5 | 1 | 40 | 540 | 2880 | 5040 | 8501 | |||
| 6 | 1 | 60 | 1320 | 13200 | 59400 | 95040 | 169021 | ||
| 7 | 1 | 84 | 2730 | 43680 | 360360 | 1441440 | 2162160 | 4010455 | |
| 8 | 1 | 112 | 5040 | 117600 | 1528800 | 11007360 | 40360320 | 57657600 | 110676833 |
All 15 rows b-files: A089503 ?
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Technical details
- Node id
n-ee693a689121a12c· revision 0 · JSON- Definition source
oeis_catalog:A089503- Fingerprint
f6-fe953b9def4ff3da
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 -1)
- 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 ?
Row sums ? A251568
Submission text ?
Atlas rows n = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 1, 4; 3: 1, 12, 30; 4: 1, 24, 168, 336; 5: 1, 40, 540, 2880, 5040; 6: 1, 60, 1320, 13200, 59400, 95040; 7: 1, 84, 2730, 43680, 360360, 1441440, 2162160; 8: 1, 112, 5040, 117600, 1528800, 11007360, 40360320, 57657600; 9: 1, 144, 8568, 274176, 5140800, 57576960, 374250240, 1283143680, 1764322560; 10: 1, 180, 13680, 574560, 14651280, 234420480, 2344204800, 14065228800, 45711993600, 60949324800;
%F FormulaThe row polynomials satisfy P(n,x) = (1+(4*n-4)*x+(n^2-3*n+2)*x^2)*P(n-1,x) - (x^2+(n-1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n - 4)*x + (n^2 - 3*n + 2)*x^2)*P[n - 1, x] - (x^2 + (n - 1)*x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (4*n - 4)*x + (n^2 - 3*n + 2)*x^2)*v[i - 1] - (x^2 + (n - 1)*x^3)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
A089503 triangle, offset 1, from P1
%e ExampleTriangle begins: 1: 1; 2: 1, 4; 3: 1, 12, 30; 4: 1, 24, 168, 336; 5: 1, 40, 540, 2880, 5040; 6: 1, 60, 1320, 13200, 59400, 95040; 7: 1, 84, 2730, 43680, 360360, 1441440, 2162160; 8: 1, 112, 5040, 117600, 1528800, 11007360, 40360320, 57657600; 9: 1, 144, 8568, 274176, 5140800, 57576960, 374250240, 1283143680, 1764322560; 10: 1, 180, 13680, 574560, 14651280, 234420480, 2344204800, 14065228800, 45711993600, 60949324800;
%F FormulaThe row polynomials satisfy P(n,x) = (1+(4*n-4)*x+(n^2-3*n+2)*x^2)*P(n-1,x) - (x^2+(n-1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n - 4)*x + (n^2 - 3*n + 2)*x^2)*P[n - 1, x] - (x^2 + (n - 1)*x^3)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (4*n - 4)*x + (n^2 - 3*n + 2)*x^2)*v[i - 1] - (x^2 + (n - 1)*x^3)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); 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
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0)
-
Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences)
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences)
Proof chains ? JSON
-
Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
Used in 7
- shift_x_plus_one of Triangle of numbers used for basis change between certain f… Image under a real-rootedness preserver (affine, shift 0)
- derivative of Triangle of numbers used for basis change between certain falling… Image under a real-rootedness preserver (derivative polynomial, shift 0)
- one_plus_derivative of Triangle of numbers used for basis change between certai… Image under a real-rootedness preserver (derivative polynomial, shift 0)
- reversal of Triangle of numbers used for basis change between certain falling f… Image under a real-rootedness preserver (reversal, shift 0)
- theta_plus_one_squared of Triangle of numbers used for basis change between cer… Image under a real-rootedness preserver (euler polynomial, shift 0)
- mobius_x_over_x_plus_one of Triangle of numbers used for basis change between c… Image under a real-rootedness preserver (mobius, shift 0)
- theta_plus_one of Triangle of numbers used for basis change between certain fal… Image under a real-rootedness preserver (euler polynomial, shift 0)
Related ?
Same fingerprint ?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.