A132166 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in [−37, −1)coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(3 + n\right)\,P_{n}(x) = \left(50 + 2\,x + 6\,n\right)\,P_{n-1}(x) + \left(37 + 38\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 379 + 44\,x + x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 379 | 44 | 1 | 424 | |||||||
| 1 | 5713 | 1242 | 66 | 1 | 7022 | ||||||
| 2 | 80032 | 28102 | 2589 | 88 | 1 | 110812 | |||||
| 3 | 1080325 | 555077 | 77815 | 4420 | 110 | 1 | 1717748 | ||||
| 4 | 14354557 | 10012512 | 1975443 | 165500 | 6735 | 132 | 1 | 26514880 | |||
| 5 | 189853438 | 169546039 | 44572143 | 5125694 | 301805 | 9534 | 154 | 1 | 409408808 | ||
| 6 | 2511177679 | 2744639366 | 922030813 | 138677168 | 11024650 | 497378 | 12817 | 176 | 1 | 6328060048 | |
| 7 | 33258000301 | 43005575310 | 17852826033 | 3388726806 | 347541537 | 20925387 | 762867 | 16584 | 198 | 1 | 97874375024 |
Compute more
?
Technical details
- Node id
n-6cf51bee0786f309· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-67df8a2168740ae3
Observed on the stored rows ?
- Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
- 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 ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 379, 44, 1; 1: 5713, 1242, 66, 1; 2: 80032, 28102, 2589, 88, 1; 3: 1080325, 555077, 77815, 4420, 110, 1; 4: 14354557, 10012512, 1975443, 165500, 6735, 132, 1; 5: 189853438, 169546039, 44572143, 5125694, 301805, 9534, 154, 1; 6: 2511177679, 2744639366, 922030813, 138677168, 11024650, 497378, 12817, 176, 1; 7: 33258000301, 43005575310, 17852826033, 3388726806, 347541537, 20925387, 762867, 16584, 198, 1; 8: 440953665088, 657853033564, 328323375558, 76477957680, 9790928625, 752828832, 36315237, 1108920, 20835, 220, 1; 9: 5849978617585, 9882747323963, 5796735000301, 1619545476081, 252345070860, 24011602581, 1467805899, 58915788, 1546185, 25570, 242, 1;
%F FormulaThe row polynomials satisfy (n+3)*P(n,x) = (6*n+50+2*x)*P(n-1,x) + (37+38*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 379 + 44*x + x^2.
%t MathematicaP[0, x_] := 379 + 44*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 50 + 2*x)*P[n - 1, x] + (37 + 38*x + x^2)*D[P[n - 1, x], x])/(n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 379 + 44*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 50 + 2*x)*v[i - 1] + (37 + 38*x + x^2)*deriv(v[i - 1], x))/(n + 3)); 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
Properties
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-37, -1)
- 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, unordered) for every m ≥ 1
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
A convolution triangle of numbers obtained from A036224.?
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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
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from
A convolution triangle of numbers obtained from A036224.?
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