A281000 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(4\,n - 2\,n x + 8\,n^{2} + 10\,n^{2} x + 18\,n^{3} x + n^{3} x^{2} - 20\,n^{4} x - 3\,n^{5} x^{2} + 2\,n^{6} x^{2}\right)\,P_{n}(x) = \left(-8 + 36\,x + 36\,x^{2} + 2\,x^{3} - 156\,n x - 110\,n x^{2} - 7\,n x^{3} + 32\,n^{2} + 100\,n^{2} x + 24\,n^{2} x^{2} + 3\,n^{2} x^{3} + 112\,n^{3} x + 160\,n^{3} x^{2} + 12\,n^{3} x^{3} - 80\,n^{4} x - 102\,n^{4} x^{2} - 12\,n^{4} x^{3} - 16\,n^{5} x^{2} - 4\,n^{5} x^{3} + 8\,n^{6} x^{2} + 6\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-16\,x - 36\,x^{2} + 6\,x^{3} + x^{4} + 64\,n x + 88\,n x^{2} - 14\,n x^{3} - 3\,n x^{4} + 24\,n^{2} x^{2} - 6\,n^{2} x^{3} - 160\,n^{3} x^{2} + 16\,n^{3} x^{3} + 6\,n^{3} x^{4} + 12\,n^{4} x^{3} - 3\,n^{4} x^{4} + 16\,n^{5} x^{3} - 4\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 3 | 4 | 7 | ||||||
| 3 | 10 | 40 | 9 | 59 | |||||
| 4 | 35 | 280 | 189 | 16 | 520 | ||||
| 5 | 126 | 1680 | 2268 | 576 | 25 | 4675 | |||
| 6 | 462 | 9240 | 20790 | 10560 | 1375 | 36 | 42463 | ||
| 7 | 1716 | 48048 | 162162 | 137280 | 35750 | 2808 | 49 | 387813 | |
| 8 | 6435 | 240240 | 1135134 | 1441440 | 625625 | 98280 | 5145 | 64 | 3552363 |
Compute more
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Technical details
- Node id
n-8b1a9aa04967006b· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-aeca54f94370d2fd
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: 3, 4; 3: 10, 40, 9; 4: 35, 280, 189, 16; 5: 126, 1680, 2268, 576, 25; 6: 462, 9240, 20790, 10560, 1375, 36; 7: 1716, 48048, 162162, 137280, 35750, 2808, 49; 8: 6435, 240240, 1135134, 1441440, 625625, 98280, 5145, 64; 9: 24310, 1166880, 7351344, 13069056, 8508500, 2227680, 233240, 8704, 81; 10: 92378, 5542680, 44895708, 106419456, 96996900, 38093328, 6647340, 496128, 13851, 100;
%F FormulaThe row polynomials satisfy (8*n^2+4*n-(20*n^4-18*n^3-10*n^2+2*n)*x+(2*n^6-3*n^5+n^3)*x^2)*P(n,x) = (32*n^2-8-(80*n^4-112*n^3-100*n^2+156*n-36)*x+(8*n^6-16*n^5-102*n^4+160*n^3+24*n^2-110*n+36)*x^2+(6*n^6-4*n^5-12*n^4+12*n^3+3*n^2-7*n+2)*x^3)*P(n-1,x) + ((64*n-16)*x-(160*n^3-24*n^2-88*n+36)*x^2+(16*n^5+12*n^4+16*n^3-6*n^2-14*n+6)*x^3-(4*n^5+3*n^4-6*n^3+3*n-1)*x^4)*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[((32*n^2 - 8 - (80*n^4 - 112*n^3 - 100*n^2 + 156*n - 36)*x + (8*n^6 - 16*n^5 - 102*n^4 + 160*n^3 + 24*n^2 - 110*n + 36)*x^2 + (6*n^6 - 4*n^5 - 12*n^4 + 12*n^3 + 3*n^2 - 7*n + 2)*x^3)*P[n - 1, x] + ((64*n - 16)*x - (160*n^3 - 24*n^2 - 88*n + 36)*x^2 + (16*n^5 + 12*n^4 + 16*n^3 - 6*n^2 - 14*n + 6)*x^3 - (4*n^5 + 3*n^4 - 6*n^3 + 3*n - 1)*x^4)*D[P[n - 1, x], x])/(8*n^2 + 4*n - (20*n^4 - 18*n^3 - 10*n^2 + 2*n)*x + (2*n^6 - 3*n^5 + n^3)*x^2)]];
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] = ((32*n^2 - 8 - (80*n^4 - 112*n^3 - 100*n^2 + 156*n - 36)*x + (8*n^6 - 16*n^5 - 102*n^4 + 160*n^3 + 24*n^2 - 110*n + 36)*x^2 + (6*n^6 - 4*n^5 - 12*n^4 + 12*n^3 + 3*n^2 - 7*n + 2)*x^3)*v[i - 1] + ((64*n - 16)*x - (160*n^3 - 24*n^2 - 88*n + 36)*x^2 + (16*n^5 + 12*n^4 + 16*n^3 - 6*n^2 - 14*n + 6)*x^3 - (4*n^5 + 3*n^4 - 6*n^3 + 3*n - 1)*x^4)*deriv(v[i - 1], x))/(8*n^2 + 4*n - (20*n^4 - 18*n^3 - 10*n^2 + 2*n)*x + (2*n^6 - 3*n^5 + 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, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
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Coefficients of one sign?
Every nonzero row, n ≥ 1: 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 ≥ 1, 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
Triangle read by rows: T(n,k) = binomial(2*n+1, 2*k+1)*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+1, 2*k+1)*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.