A133214 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(2 - 48\,x + 128\,x^{2} + n - 80\,n x + 320\,n x^{2} - 44\,n^{2} x + 320\,n^{2} x^{2} - 8\,n^{3} x + 160\,n^{3} x^{2} + 40\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(2 - 16\,x - 440\,x^{2} + 1344\,x^{3} + n - 56\,n x - 644\,n x^{2} + 3232\,n x^{3} - 44\,n^{2} x - 192\,n^{2} x^{2} + 2992\,n^{2} x^{3} - 8\,n^{3} x + 76\,n^{3} x^{2} + 1336\,n^{3} x^{3} + 40\,n^{4} x^{2} + 288\,n^{4} x^{3} + 4\,n^{5} x^{2} + 24\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(2\,x - 96\,x^{2} + 568\,x^{3} - 768\,x^{4} - 80\,n x^{2} + 768\,n x^{3} - 1216\,n x^{4} - 16\,n^{2} x^{2} + 384\,n^{2} x^{3} - 704\,n^{2} x^{4} + 88\,n^{3} x^{3} - 176\,n^{3} x^{4} + 8\,n^{4} x^{3} - 16\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 32\,x + 36\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 32 | 36 | 69 | |||||||
| 1 | 1 | 72 | 324 | 128 | 525 | ||||||
| 2 | 1 | 128 | 1296 | 2048 | 400 | 3873 | |||||
| 3 | 1 | 200 | 3600 | 12800 | 10000 | 1152 | 27753 | ||||
| 4 | 1 | 288 | 8100 | 51200 | 90000 | 41472 | 3136 | 194197 | |||
| 5 | 1 | 392 | 15876 | 156800 | 490000 | 508032 | 153664 | 8192 | 1332957 | ||
| 6 | 1 | 512 | 28224 | 401408 | 1960000 | 3612672 | 2458624 | 524288 | 20736 | 9006465 | |
| 7 | 1 | 648 | 46656 | 903168 | 6350400 | 18289152 | 22127616 | 10616832 | 1679616 | 51200 | 60065289 |
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Technical details
- Node id
n-51f0a51cbdea3138· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-79c4b9a4e04244e4
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: 1, 32, 36; 1: 1, 72, 324, 128; 2: 1, 128, 1296, 2048, 400; 3: 1, 200, 3600, 12800, 10000, 1152; 4: 1, 288, 8100, 51200, 90000, 41472, 3136; 5: 1, 392, 15876, 156800, 490000, 508032, 153664, 8192; 6: 1, 512, 28224, 401408, 1960000, 3612672, 2458624, 524288, 20736; 7: 1, 648, 46656, 903168, 6350400, 18289152, 22127616, 10616832, 1679616, 51200; 8: 1, 800, 72900, 1843200, 17640000, 73156608, 138297600, 117964800, 41990400, 5120000, 123904; 9: 1, 968, 108900, 3484800, 43560000, 245887488, 669360384, 892108800, 564537600, 154880000, 14992384, 294912;
%F FormulaThe row polynomials satisfy (n+2-(8*n^3+44*n^2+80*n+48)*x+(4*n^5+40*n^4+160*n^3+320*n^2+320*n+128)*x^2)*P(n,x) = (n+2-(8*n^3+44*n^2+56*n+16)*x+(4*n^5+40*n^4+76*n^3-192*n^2-644*n-440)*x^2+(24*n^5+288*n^4+1336*n^3+2992*n^2+3232*n+1344)*x^3)*P(n-1,x) + (2*x-(16*n^2+80*n+96)*x^2+(8*n^4+88*n^3+384*n^2+768*n+568)*x^3-(16*n^4+176*n^3+704*n^2+1216*n+768)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 32*x + 36*x^2.
%t MathematicaP[0, x_] := 1 + 32*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 2 - (8*n^3 + 44*n^2 + 56*n + 16)*x + (4*n^5 + 40*n^4 + 76*n^3 - 192*n^2 - 644*n - 440)*x^2 + (24*n^5 + 288*n^4 + 1336*n^3 + 2992*n^2 + 3232*n + 1344)*x^3)*P[n - 1, x] + (2*x - (16*n^2 + 80*n + 96)*x^2 + (8*n^4 + 88*n^3 + 384*n^2 + 768*n + 568)*x^3 - (16*n^4 + 176*n^3 + 704*n^2 + 1216*n + 768)*x^4)*D[P[n - 1, x], x])/(n + 2 - (8*n^3 + 44*n^2 + 80*n + 48)*x + (4*n^5 + 40*n^4 + 160*n^3 + 320*n^2 + 320*n + 128)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1 + 32*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 2 - (8*n^3 + 44*n^2 + 56*n + 16)*x + (4*n^5 + 40*n^4 + 76*n^3 - 192*n^2 - 644*n - 440)*x^2 + (24*n^5 + 288*n^4 + 1336*n^3 + 2992*n^2 + 3232*n + 1344)*x^3)*v[i - 1] + (2*x - (16*n^2 + 80*n + 96)*x^2 + (8*n^4 + 88*n^3 + 384*n^2 + 768*n + 568)*x^3 - (16*n^4 + 176*n^3 + 704*n^2 + 1216*n + 768)*x^4)*deriv(v[i - 1], x))/(n + 2 - (8*n^3 + 44*n^2 + 80*n + 48)*x + (4*n^5 + 40*n^4 + 160*n^3 + 320*n^2 + 320*n + 128)*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 (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
Delannoy paths counted by number of weak peaks.?
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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
Delannoy paths counted by number of weak peaks.?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
- [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.
- [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023