A367270 → basis A013610 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(-n + 3\,n x + 5\,n^{2} - 21\,n^{2} x - 8\,n^{3} + 54\,n^{3} x - 9\,n^{3} x^{2} + 4\,n^{4} - 54\,n^{4} x + 27\,n^{4} x^{2} + 18\,n^{5} x - 27\,n^{5} x^{2} + 9\,n^{6} x^{2}\right)\,P_{n}(x) = \left(2 + 18\,x + 54\,x^{2} + 54\,x^{3} - 14\,n - 102\,n x - 243\,n x^{2} - 135\,n x^{3} + 36\,n^{2} + 180\,n^{2} x + 198\,n^{2} x^{2} - 40\,n^{3} - 60\,n^{3} x + 306\,n^{3} x^{2} + 243\,n^{3} x^{3} + 16\,n^{4} - 108\,n^{4} x - 495\,n^{4} x^{2} - 162\,n^{4} x^{3} + 72\,n^{5} x + 144\,n^{5} x^{2} - 81\,n^{5} x^{3} + 36\,n^{6} x^{2} + 81\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(x + 9\,x^{2} + 27\,x^{3} + 27\,x^{4} - 6\,n x - 42\,n x^{2} - 90\,n x^{3} - 54\,n x^{4} + 12\,n^{2} x + 48\,n^{2} x^{2} + 27\,n^{2} x^{3} - 27\,n^{2} x^{4} - 8\,n^{3} x + 12\,n^{3} x^{2} + 144\,n^{3} x^{3} + 108\,n^{3} x^{4} - 36\,n^{4} x^{2} - 117\,n^{4} x^{3} - 27\,n^{4} x^{4} - 18\,n^{5} x^{3} - 54\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | Pn(1) |
|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||
| 1 | 1 | 1 | ||||||
| 2 | 3 | 24 | 27 | |||||
| 3 | 10 | 144 | 243 | 397 | ||||
| 4 | 35 | 720 | 2430 | 1728 | 4913 | |||
| 5 | 126 | 3360 | 17010 | 25920 | 10125 | 56541 | ||
| 6 | 462 | 15120 | 102060 | 241920 | 212625 | 52488 | 624675 | |
| 7 | 1716 | 66528 | 561330 | 1814400 | 2551500 | 1469664 | 250047 | 6715185 |
Compute more
?
Technical details
- Node id
n-832c931ce4d0668a· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-9c4fb9d58fe9065f
Observed on the stored rows ?
- Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
- 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: 3, 24; 3: 10, 144, 243; 4: 35, 720, 2430, 1728; 5: 126, 3360, 17010, 25920, 10125; 6: 462, 15120, 102060, 241920, 212625, 52488; 7: 1716, 66528, 561330, 1814400, 2551500, 1469664, 250047; 8: 6435, 288288, 2918916, 11975040, 23388750, 22044960, 9001692, 1119744; 9: 24310, 1235520, 14594580, 72648576, 182432250, 242494560, 165031020, 50388480, 4782969;
%F FormulaThe row polynomials satisfy (4*n^4-8*n^3+5*n^2-n+(18*n^5-54*n^4+54*n^3-21*n^2+3*n)*x+(9*n^6-27*n^5+27*n^4-9*n^3)*x^2)*P(n,x) = (16*n^4-40*n^3+36*n^2-14*n+2+(72*n^5-108*n^4-60*n^3+180*n^2-102*n+18)*x+(36*n^6+144*n^5-495*n^4+306*n^3+198*n^2-243*n+54)*x^2+(81*n^6-81*n^5-162*n^4+243*n^3-135*n+54)*x^3)*P(n-1,x) - ((8*n^3-12*n^2+6*n-1)*x+(36*n^4-12*n^3-48*n^2+42*n-9)*x^2+(18*n^5+117*n^4-144*n^3-27*n^2+90*n-27)*x^3+(54*n^5+27*n^4-108*n^3+27*n^2+54*n-27)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^4 - 40*n^3 + 36*n^2 - 14*n + 2 + (72*n^5 - 108*n^4 - 60*n^3 + 180*n^2 - 102*n + 18)*x + (36*n^6 + 144*n^5 - 495*n^4 + 306*n^3 + 198*n^2 - 243*n + 54)*x^2 + (81*n^6 - 81*n^5 - 162*n^4 + 243*n^3 - 135*n + 54)*x^3)*P[n - 1, x] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (36*n^4 - 12*n^3 - 48*n^2 + 42*n - 9)*x^2 + (18*n^5 + 117*n^4 - 144*n^3 - 27*n^2 + 90*n - 27)*x^3 + (54*n^5 + 27*n^4 - 108*n^3 + 27*n^2 + 54*n - 27)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 8*n^3 + 5*n^2 - n + (18*n^5 - 54*n^4 + 54*n^3 - 21*n^2 + 3*n)*x + (9*n^6 - 27*n^5 + 27*n^4 - 9*n^3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^4 - 40*n^3 + 36*n^2 - 14*n + 2 + (72*n^5 - 108*n^4 - 60*n^3 + 180*n^2 - 102*n + 18)*x + (36*n^6 + 144*n^5 - 495*n^4 + 306*n^3 + 198*n^2 - 243*n + 54)*x^2 + (81*n^6 - 81*n^5 - 162*n^4 + 243*n^3 - 135*n + 54)*x^3)*v[i - 1] - ((8*n^3 - 12*n^2 + 6*n - 1)*x + (36*n^4 - 12*n^3 - 48*n^2 + 42*n - 9)*x^2 + (18*n^5 + 117*n^4 - 144*n^3 - 27*n^2 + 90*n - 27)*x^3 + (54*n^5 + 27*n^4 - 108*n^3 + 27*n^2 + 54*n - 27)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 8*n^3 + 5*n^2 - n + (18*n^5 - 54*n^4 + 54*n^3 - 21*n^2 + 3*n)*x + (9*n^6 - 27*n^5 + 27*n^4 - 9*n^3)*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 ≥ 2
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
basis_map_A013610 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
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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(n, k)*binomial(n - 1, n - k - 1).?
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from
basis_map_A013610 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
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