A330798 → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(2\,n^{2} + 4\,n^{2} x + 2\,n^{2} x^{2} + 4\,n^{3} + 8\,n^{3} x + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-2\,x - 12\,x^{2} - 18\,x^{3} - 8\,x^{4} - 7\,n - 22\,n x - 3\,n x^{2} + 32\,n x^{3} + 20\,n x^{4} + 6\,n^{2} + 26\,n^{2} x + 42\,n^{2} x^{2} + 30\,n^{2} x^{3} + 8\,n^{2} x^{4} + 13\,n^{3} + 46\,n^{3} x + 33\,n^{3} x^{2} - 20\,n^{3} x^{3} - 20\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-1 - 9\,x - 29\,x^{2} - 43\,x^{3} - 30\,x^{4} - 8\,x^{5} + 6\,n x + 30\,n x^{2} + 54\,n x^{3} + 42\,n x^{4} + 12\,n x^{5} + n^{2} + 15\,n^{2} x + 59\,n^{2} x^{2} + 97\,n^{2} x^{3} + 72\,n^{2} x^{4} + 20\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 4 | 6 | ||||||
| 2 | 6 | 27 | 30 | 63 | |||||
| 3 | 20 | 144 | 340 | 264 | 768 | ||||
| 4 | 70 | 700 | 2580 | 4165 | 2490 | 10005 | |||
| 5 | 252 | 3240 | 16380 | 40800 | 50166 | 24396 | 135234 | ||
| 6 | 924 | 14553 | 93940 | 318780 | 600810 | 597135 | 244790 | 1870932 | |
| 7 | 3432 | 64064 | 504504 | 2176720 | 5565560 | 8443680 | 7045480 | 2496528 | 26299968 |
Compute more
?
Technical details
- Node id
n-b11137933d9f04e2· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-b24c47ab26a256b2
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 ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 2, 4; 2: 6, 27, 30; 3: 20, 144, 340, 264; 4: 70, 700, 2580, 4165, 2490; 5: 252, 3240, 16380, 40800, 50166, 24396; 6: 924, 14553, 93940, 318780, 600810, 597135, 244790; 7: 3432, 64064, 504504, 2176720, 5565560, 8443680, 7045480, 2496528; 8: 12870, 277992, 2588040, 13585572, 44040780, 90384840, 114794568, 82561545, 25763058; 9: 48620, 1193400, 12835680, 79511040, 312989040, 812760480, 1393513800, 1522350720, 962205610, 268243860;
%F FormulaThe row polynomials satisfy (4*n^3+2*n^2+(8*n^3+4*n^2)*x+(4*n^3+2*n^2)*x^2)*P(n,x) = (13*n^3+6*n^2-7*n+(46*n^3+26*n^2-22*n-2)*x+(33*n^3+42*n^2-3*n-12)*x^2-(20*n^3-30*n^2-32*n+18)*x^3-(20*n^3-8*n^2-20*n+8)*x^4)*P(n-1,x) + (n^2-1+(15*n^2+6*n-9)*x+(59*n^2+30*n-29)*x^2+(97*n^2+54*n-43)*x^3+(72*n^2+42*n-30)*x^4+(20*n^2+12*n-8)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((13*n^3 + 6*n^2 - 7*n + (46*n^3 + 26*n^2 - 22*n - 2)*x + (33*n^3 + 42*n^2 - 3*n - 12)*x^2 - (20*n^3 - 30*n^2 - 32*n + 18)*x^3 - (20*n^3 - 8*n^2 - 20*n + 8)*x^4)*P[n - 1, x] + (n^2 - 1 + (15*n^2 + 6*n - 9)*x + (59*n^2 + 30*n - 29)*x^2 + (97*n^2 + 54*n - 43)*x^3 + (72*n^2 + 42*n - 30)*x^4 + (20*n^2 + 12*n - 8)*x^5)*D[P[n - 1, x], x])/(4*n^3 + 2*n^2 + (8*n^3 + 4*n^2)*x + (4*n^3 + 2*n^2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((13*n^3 + 6*n^2 - 7*n + (46*n^3 + 26*n^2 - 22*n - 2)*x + (33*n^3 + 42*n^2 - 3*n - 12)*x^2 - (20*n^3 - 30*n^2 - 32*n + 18)*x^3 - (20*n^3 - 8*n^2 - 20*n + 8)*x^4)*v[i - 1] + (n^2 - 1 + (15*n^2 + 6*n - 9)*x + (59*n^2 + 30*n - 29)*x^2 + (97*n^2 + 54*n - 43)*x^3 + (72*n^2 + 42*n - 30)*x^4 + (20*n^2 + 12*n - 8)*x^5)*deriv(v[i - 1], x))/(4*n^3 + 2*n^2 + (8*n^3 + 4*n^2)*x + (4*n^3 + 2*n^2)*x^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
Properties
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 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)
Proof chains ? JSON
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows, interpolating between the central binomial coefficients and the central coefficients of the Cata…?
-
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
-
-
from
Triangle read by rows, interpolating between the central binomial coefficients and the central coefficients of the Cata…?
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.