A243662 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(2\,n - 2\,n x - 8\,n^{2} + 10\,n^{2} x - 6\,n^{3} - 6\,n^{3} x + 2\,n^{3} x^{2} + 36\,n^{4} - 50\,n^{4} x + 36\,n^{5} x - 6\,n^{5} x^{2} + 4\,n^{6} x^{2}\right)\,P_{n}(x) = \left(6 + 10\,x + 12\,x^{2} + 4\,x^{3} - 63\,n - 91\,n x - 86\,n x^{2} - 14\,n x^{3} + 243\,n^{2} + 258\,n^{2} x + 134\,n^{2} x^{2} + 6\,n^{2} x^{3} - 405\,n^{3} - 162\,n^{3} x + 87\,n^{3} x^{2} + 24\,n^{3} x^{3} + 243\,n^{4} - 270\,n^{4} x - 273\,n^{4} x^{2} - 24\,n^{4} x^{3} + 243\,n^{5} x + 99\,n^{5} x^{2} - 8\,n^{5} x^{3} + 27\,n^{6} x^{2} + 12\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 5\,x^{2} + 6\,x^{3} + 2\,x^{4} - 27\,n x - 38\,n x^{2} - 32\,n x^{3} - 6\,n x^{4} + 81\,n^{2} x + 72\,n^{2} x^{2} + 31\,n^{2} x^{3} - 81\,n^{3} x + 55\,n^{3} x^{3} + 12\,n^{3} x^{4} - 81\,n^{4} x^{2} - 75\,n^{4} x^{3} - 6\,n^{4} x^{4} - 9\,n^{5} x^{3} - 8\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 3 | 4 | 7 | ||||||
| 3 | 12 | 32 | 9 | 53 | |||||
| 4 | 55 | 220 | 135 | 16 | 426 | ||||
| 5 | 273 | 1456 | 1404 | 384 | 25 | 3542 | |||
| 6 | 1428 | 9520 | 12600 | 5600 | 875 | 36 | 30059 | ||
| 7 | 7752 | 62016 | 104652 | 65280 | 17000 | 1728 | 49 | 258477 | |
| 8 | 43263 | 403788 | 829521 | 670320 | 249375 | 43092 | 3087 | 64 | 2242510 |
Compute more
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Technical details
- Node id
n-199db35c0fa198cb· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-3a4012a76922fd57
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: 12, 32, 9; 4: 55, 220, 135, 16; 5: 273, 1456, 1404, 384, 25; 6: 1428, 9520, 12600, 5600, 875, 36; 7: 7752, 62016, 104652, 65280, 17000, 1728, 49; 8: 43263, 403788, 829521, 670320, 249375, 43092, 3087, 64; 9: 246675, 2631200, 6375600, 6347264, 3099250, 776160, 96040, 5120, 81; 10: 1430715, 17168580, 47953620, 56833920, 34534500, 11476080, 2082696, 194304, 8019, 100;
%F FormulaThe row polynomials satisfy (36*n^4-6*n^3-8*n^2+2*n+(36*n^5-50*n^4-6*n^3+10*n^2-2*n)*x+(4*n^6-6*n^5+2*n^3)*x^2)*P(n,x) = (243*n^4-405*n^3+243*n^2-63*n+6+(243*n^5-270*n^4-162*n^3+258*n^2-91*n+10)*x+(27*n^6+99*n^5-273*n^4+87*n^3+134*n^2-86*n+12)*x^2+(12*n^6-8*n^5-24*n^4+24*n^3+6*n^2-14*n+4)*x^3)*P(n-1,x) - ((81*n^3-81*n^2+27*n-3)*x+(81*n^4-72*n^2+38*n-5)*x^2+(9*n^5+75*n^4-55*n^3-31*n^2+32*n-6)*x^3+(8*n^5+6*n^4-12*n^3+6*n-2)*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[((243*n^4 - 405*n^3 + 243*n^2 - 63*n + 6 + (243*n^5 - 270*n^4 - 162*n^3 + 258*n^2 - 91*n + 10)*x + (27*n^6 + 99*n^5 - 273*n^4 + 87*n^3 + 134*n^2 - 86*n + 12)*x^2 + (12*n^6 - 8*n^5 - 24*n^4 + 24*n^3 + 6*n^2 - 14*n + 4)*x^3)*P[n - 1, x] - ((81*n^3 - 81*n^2 + 27*n - 3)*x + (81*n^4 - 72*n^2 + 38*n - 5)*x^2 + (9*n^5 + 75*n^4 - 55*n^3 - 31*n^2 + 32*n - 6)*x^3 + (8*n^5 + 6*n^4 - 12*n^3 + 6*n - 2)*x^4)*D[P[n - 1, x], x])/(36*n^4 - 6*n^3 - 8*n^2 + 2*n + (36*n^5 - 50*n^4 - 6*n^3 + 10*n^2 - 2*n)*x + (4*n^6 - 6*n^5 + 2*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] = ((243*n^4 - 405*n^3 + 243*n^2 - 63*n + 6 + (243*n^5 - 270*n^4 - 162*n^3 + 258*n^2 - 91*n + 10)*x + (27*n^6 + 99*n^5 - 273*n^4 + 87*n^3 + 134*n^2 - 86*n + 12)*x^2 + (12*n^6 - 8*n^5 - 24*n^4 + 24*n^3 + 6*n^2 - 14*n + 4)*x^3)*v[i - 1] - ((81*n^3 - 81*n^2 + 27*n - 3)*x + (81*n^4 - 72*n^2 + 38*n - 5)*x^2 + (9*n^5 + 75*n^4 - 55*n^3 - 31*n^2 + 32*n - 6)*x^3 + (8*n^5 + 6*n^4 - 12*n^3 + 6*n - 2)*x^4)*deriv(v[i - 1], x))/(36*n^4 - 6*n^3 - 8*n^2 + 2*n + (36*n^5 - 50*n^4 - 6*n^3 + 10*n^2 - 2*n)*x + (4*n^6 - 6*n^5 + 2*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
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: the reversed x = 1+q Narayana triangle at m=2.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.?
- ↑ Iterated real-rootedness preservers (above)
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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: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).?
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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) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).?
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from
Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.?
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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 T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.?
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.?
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.