basis-map coefficients of A021012 → basis A008277 → x ↦ x+1
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(2 + x - 2\,n - n x\right)\,P_{n-1}(x) + \left(1 + x - n - n x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | -2 | -1 | -3 | |||||||||||||
| 3 | 10 | 10 | 2 | 22 | ||||||||||||
| 4 | -90 | -132 | -54 | -6 | -282 | |||||||||||
| 5 | 1248 | 2376 | 1464 | 336 | 24 | 5448 | ||||||||||
| 6 | -24360 | -56520 | -46200 | -16200 | -2400 | -120 | -145800 | |||||||||
| 7 | 631440 | 1717920 | 1739520 | 820800 | 187200 | 19440 | 720 | 5117040 | ||||||||
| 8 | -20865600 | -64849680 | -77968800 | -46146240 | -14288400 | -2293200 | -176400 | -5040 | -226593360 | |||||||
| 9 | 852647040 | 2970817920 | 4121308800 | 2926828800 | 1146741120 | 251193600 | 29917440 | 1774080 | 40320 | 12301269120 | ||||||
| 10 | -42085008000 | -162069465600 | -254128855680 | -210081755520 | -99569191680 | -27761408640 | -4526565120 | -415860480 | -19595520 | -362880 | -800658069120 | |||||
| 11 | 2462394816000 | 10365511161600 | 18088301548800 | 17028144000000 | 9463039488000 | 3210584428800 | 668849529600 | 84260736000 | 6150816000 | 235872000 | 3628800 | 61377476025600 | ||||
| 12 | -168193308729600 | -767090845382400 | -1471834642924800 | -1551892974508800 | -988452333792000 | -395452504339200 | -100663263993600 | -16240429497600 | -1626809184000 | -96598656000 | -3073593600 | -39916800 | -5461546824518400 | |||
| 13 | 13241729554099200 | 64957621568716800 | 135721300087296000 | 158221306207641600 | 113518433987481600 | 52327193377536000 | 15773299473254400 | 3118099235328000 | 400534910899200 | 32641564032000 | 1607050368000 | 43110144000 | 479001600 | 557313810605798400 | ||
| 14 | -1188734048799897600 | -6234243527652940800 | -14072587627030732800 | -17951720371980595200 | -14312582401262745600 | -7467831598106880000 | -2604423689541273600 | -611526134419200000 | -96423891467904000 | -10083614046105600 | -681204940416000 | -28251993369600 | -647610163200 | -6227020800 | -64550867015079244800 | |
| 15 | 120563962753538304000 | 672512958401482598400 | 1629316572124047206400 | 2255141267287785676800 | 1976329218781280563200 | 1150995240153619660800 | 456174656775200563200 | 124313700434802278400 | 23331246057184665600 | 2998179735309158400 | 259963833614284800 | 14787530466508800 | 523592816947200 | 10374216652800 | 87178291200 | 8411952287922543360000 |
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Technical details
- Node id
n-3e5803db2c95b6c4· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-1090d35600282d8c
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): no - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: -2, -1; 3: 10, 10, 2; 4: -90, -132, -54, -6; 5: 1248, 2376, 1464, 336, 24; 6: -24360, -56520, -46200, -16200, -2400, -120; 7: 631440, 1717920, 1739520, 820800, 187200, 19440, 720; 8: -20865600, -64849680, -77968800, -46146240, -14288400, -2293200, -176400, -5040; 9: 852647040, 2970817920, 4121308800, 2926828800, 1146741120, 251193600, 29917440, 1774080, 40320; 10: -42085008000, -162069465600, -254128855680, -210081755520, -99569191680, -27761408640, -4526565120, -415860480, -19595520, -362880;
%F FormulaThe row polynomials satisfy P(n,x) = -(2*n-2+(n-1)*x)*P(n-1,x) - (n-1+(n-1)*x)*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[-(2*n - 2 + (n - 1)*x)*P[n - 1, x] - (n - 1 + (n - 1)*x)*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] = -(2*n - 2 + (n - 1)*x)*v[i - 1] - (n - 1 + (n - 1)*x)*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
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, -1]
- Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
basis_map_A008277 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polyno…?
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Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
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from
basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
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Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
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from
basis_map_A008277 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polyno…?
Related ?
Same fingerprint ?
- basis-map coefficients of A112227 → basis A008277 → x ↦ x+1
- A008278 → x ↦ x/(x+1)
- A137597 → reverse
- A137597: Triangle read by rows: A008277 * A007318.
- A008277 → x ↦ x+1
- A130123 → basis A008277 → x ↦ x+1
- A008277 → x ↦ x+1 → reverse
References
- [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741