A108426 → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3} + 4\,n^{2} + 12\,n^{2} x + 12\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n}(x) = \left(6 + 32\,x + 68\,x^{2} + 72\,x^{3} + 38\,x^{4} + 8\,x^{5} - 27\,n - 143\,n x - 295\,n x^{2} - 297\,n x^{3} - 146\,n x^{4} - 28\,n x^{5} + 27\,n^{2} + 141\,n^{2} x + 281\,n^{2} x^{2} + 267\,n^{2} x^{3} + 120\,n^{2} x^{4} + 20\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(3\,x + 19\,x^{2} + 47\,x^{3} + 57\,x^{4} + 34\,x^{5} + 8\,x^{6} - 9\,n x - 55\,n x^{2} - 131\,n x^{3} - 153\,n x^{4} - 88\,n x^{5} - 20\,n x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 2 | 3 | ||||||
| 2 | 3 | 11 | 10 | 24 | |||||
| 3 | 12 | 64 | 113 | 66 | 255 | ||||
| 4 | 55 | 385 | 1005 | 1159 | 498 | 3102 | |||
| 5 | 273 | 2366 | 8164 | 14016 | 11969 | 4066 | 40854 | ||
| 6 | 1428 | 14756 | 63280 | 144130 | 183855 | 124515 | 34970 | 566934 | |
| 7 | 7752 | 93024 | 476748 | 1352520 | 2293640 | 2324736 | 1303777 | 312066 | 8164263 |
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Technical details
- Node id
n-346eb8fc7de6d5a4· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-a09e6eb68394f014
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: 1, 2; 2: 3, 11, 10; 3: 12, 64, 113, 66; 4: 55, 385, 1005, 1159, 498; 5: 273, 2366, 8164, 14016, 11969, 4066; 6: 1428, 14756, 63280, 144130, 183855, 124515, 34970; 7: 7752, 93024, 476748, 1352520, 2293640, 2324736, 1303777, 312066; 8: 43263, 591261, 3524367, 11966409, 25310565, 34146819, 28692153, 13726991, 2862562; 9: 246675, 3782350, 25704800, 101612896, 257471522, 433625500, 485366840, 348148480, 145198913, 26824386;
%F FormulaThe row polynomials satisfy (4*n^2+2*n+(12*n^2+6*n)*x+(12*n^2+6*n)*x^2+(4*n^2+2*n)*x^3)*P(n,x) = (27*n^2-27*n+6+(141*n^2-143*n+32)*x+(281*n^2-295*n+68)*x^2+(267*n^2-297*n+72)*x^3+(120*n^2-146*n+38)*x^4+(20*n^2-28*n+8)*x^5)*P(n-1,x) - ((9*n-3)*x+(55*n-19)*x^2+(131*n-47)*x^3+(153*n-57)*x^4+(88*n-34)*x^5+(20*n-8)*x^6)*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[((27*n^2 - 27*n + 6 + (141*n^2 - 143*n + 32)*x + (281*n^2 - 295*n + 68)*x^2 + (267*n^2 - 297*n + 72)*x^3 + (120*n^2 - 146*n + 38)*x^4 + (20*n^2 - 28*n + 8)*x^5)*P[n - 1, x] - ((9*n - 3)*x + (55*n - 19)*x^2 + (131*n - 47)*x^3 + (153*n - 57)*x^4 + (88*n - 34)*x^5 + (20*n - 8)*x^6)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (12*n^2 + 6*n)*x + (12*n^2 + 6*n)*x^2 + (4*n^2 + 2*n)*x^3)]];
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] = ((27*n^2 - 27*n + 6 + (141*n^2 - 143*n + 32)*x + (281*n^2 - 295*n + 68)*x^2 + (267*n^2 - 297*n + 72)*x^3 + (120*n^2 - 146*n + 38)*x^4 + (20*n^2 - 28*n + 8)*x^5)*v[i - 1] - ((9*n - 3)*x + (55*n - 19)*x^2 + (131*n - 47)*x^3 + (153*n - 57)*x^4 + (88*n - 34)*x^5 + (20*n - 8)*x^6)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (12*n^2 + 6*n)*x + (12*n^2 + 6*n)*x^2 + (4*n^2 + 2*n)*x^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 [-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
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch th…?
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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
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.?
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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
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from
Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.?
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from
Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch th…?
Related ?
Same fingerprint ?
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [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.