A280580 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-n + 2\,n^{2}\right)\,P_{n}(x) = \left(12 + 6\,x - 28\,n - 12\,n x + 14\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-9 - 6\,x + 3\,x^{2} + 12\,n + 8\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 2 | 1 | 3 | ||||||
| 3 | 9 | 8 | 1 | 18 | |||||
| 4 | 51 | 63 | 18 | 1 | 133 | ||||
| 5 | 323 | 508 | 230 | 32 | 1 | 1094 | |||
| 6 | 2188 | 4175 | 2590 | 610 | 50 | 1 | 9614 | ||
| 7 | 15511 | 34788 | 27405 | 9260 | 1335 | 72 | 1 | 88372 | |
| 8 | 113634 | 292845 | 279664 | 123977 | 26425 | 2569 | 98 | 1 | 839213 |
Compute more
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Technical details
- Node id
n-38f05709377aafdf· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-886b9c97c4b2ddcc
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 ?
No OEIS matches.
Submission text ?
Atlas rows n = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 2, 1; 3: 9, 8, 1; 4: 51, 63, 18, 1; 5: 323, 508, 230, 32, 1; 6: 2188, 4175, 2590, 610, 50, 1; 7: 15511, 34788, 27405, 9260, 1335, 72, 1; 8: 113634, 292845, 279664, 123977, 26425, 2569, 98, 1; 9: 853467, 2484576, 2787876, 1534512, 439250, 64456, 4508, 128, 1; 10: 6536382, 21211011, 27342360, 18007164, 6518232, 1306746, 140028, 7380, 162, 1;
%F FormulaThe row polynomials satisfy (2*n^2-n)*P(n,x) = (14*n^2-28*n+12+(6*n^2-12*n+6)*x)*P(n-1,x) + (12*n-9+(8*n-6)*x-(4*n-3)*x^2)*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[Together[((14*n^2 - 28*n + 12 + (6*n^2 - 12*n + 6)*x)*P[n - 1, x] + (12*n - 9 + (8*n - 6)*x - (4*n - 3)*x^2)*D[P[n - 1, x], x])/(2*n^2 - n)]];
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] = ((14*n^2 - 28*n + 12 + (6*n^2 - 12*n + 6)*x)*v[i - 1] + (12*n - 9 + (8*n - 6)*x - (4*n - 3)*x^2)*deriv(v[i - 1], x))/(2*n^2 - n)); 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)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
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) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=n.?
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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: T(n,k) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=n.?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.