A039755 → reverse → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(9 - 12\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-3 + 140\,x + 585\,x^{2} + 8\,n - 216\,n x - 1824\,n x^{2} - 4\,n^{2} + 144\,n^{2} x + 1976\,n^{2} x^{2} - 32\,n^{3} x - 896\,n^{3} x^{2} + 144\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 4\,x^{2} + 612\,x^{3} + 120\,n x^{2} - 1752\,n x^{3} - 112\,n^{2} x^{2} + 1664\,n^{2} x^{3} + 32\,n^{3} x^{2} - 608\,n^{3} x^{3} + 64\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(8\,x^{3} + 108\,x^{4} - 288\,n x^{4} + 240\,n^{2} x^{4} - 64\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(12 - 108\,x - 20\,n + 216\,n x + 8\,n^{2} - 144\,n^{2} x + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(4\,x + 36\,x^{2} - 120\,n x^{2} + 112\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 1 | 4 | 5 | ||||||
| 3 | 1 | 16 | 9 | 26 | |||||
| 4 | 1 | 36 | 117 | 16 | 170 | ||||
| 5 | 1 | 64 | 522 | 640 | 25 | 1252 | |||
| 6 | 1 | 100 | 1530 | 5280 | 3025 | 36 | 9972 | ||
| 7 | 1 | 144 | 3555 | 24320 | 44275 | 13104 | 49 | 85448 | |
| 8 | 1 | 196 | 7119 | 81200 | 310275 | 331884 | 53557 | 64 | 784296 |
Compute more
?
Technical details
- Node id
n-99d42afb2e5322c3· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-2bb878dfcdcd8bac
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: 1, 4; 3: 1, 16, 9; 4: 1, 36, 117, 16; 5: 1, 64, 522, 640, 25; 6: 1, 100, 1530, 5280, 3025, 36; 7: 1, 144, 3555, 24320, 44275, 13104, 49; 8: 1, 196, 7119, 81200, 310275, 331884, 53557, 64; 9: 1, 256, 12852, 220416, 1452150, 3459456, 2312212, 209920, 81; 10: 1, 324, 21492, 517440, 5240550, 22279320, 35273140, 15310080, 797121, 100;
%F FormulaThe row polynomials satisfy (4*n^2-12*n+9)*P(n,x) = -(4*n^2-8*n+3+(32*n^3-144*n^2+216*n-140)*x-(144*n^4-896*n^3+1976*n^2-1824*n+585)*x^2)*P(n-2,x) - (4*x-(32*n^3-112*n^2+120*n+4)*x^2-(64*n^4-608*n^3+1664*n^2-1752*n+612)*x^3)*d/dx P(n-2,x) + (8*x^3-(64*n^3-240*n^2+288*n-108)*x^4)*d^2/dx^2 P(n-2,x) + (8*n^2-20*n+12+(32*n^3-144*n^2+216*n-108)*x)*P(n-1,x) + (4*x-(32*n^3-112*n^2+120*n-36)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 8*n + 3 + (32*n^3 - 144*n^2 + 216*n - 140)*x - (144*n^4 - 896*n^3 + 1976*n^2 - 1824*n + 585)*x^2)*P[n - 2, x] - (4*x - (32*n^3 - 112*n^2 + 120*n + 4)*x^2 - (64*n^4 - 608*n^3 + 1664*n^2 - 1752*n + 612)*x^3)*D[P[n - 2, x], x] + (8*x^3 - (64*n^3 - 240*n^2 + 288*n - 108)*x^4)*D[P[n - 2, x], {x, 2}] + (8*n^2 - 20*n + 12 + (32*n^3 - 144*n^2 + 216*n - 108)*x)*P[n - 1, x] + (4*x - (32*n^3 - 112*n^2 + 120*n - 36)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 12*n + 9)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = (-(4*n^2 - 8*n + 3 + (32*n^3 - 144*n^2 + 216*n - 140)*x - (144*n^4 - 896*n^3 + 1976*n^2 - 1824*n + 585)*x^2)*v[i - 2] - (4*x - (32*n^3 - 112*n^2 + 120*n + 4)*x^2 - (64*n^4 - 608*n^3 + 1664*n^2 - 1752*n + 612)*x^3)*deriv(v[i - 2], x) + (8*x^3 - (64*n^3 - 240*n^2 + 288*n - 108)*x^4)*deriv(deriv(v[i - 2], x), x) + (8*n^2 - 20*n + 12 + (32*n^3 - 144*n^2 + 216*n - 108)*x)*v[i - 1] + (4*x - (32*n^3 - 112*n^2 + 120*n - 36)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 12*n + 9)); 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
reversal of Triangle of B-analogs of Stirling numbers of the second kind.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle of B-analogs of Stirling numbers of the second kind.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Iterated real-rootedness preservers (above)
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from
Triangle of B-analogs of Stirling numbers of the second kind.?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
-
-
from
reversal of Triangle of B-analogs of Stirling numbers of the second kind.?
Related ?
Same fingerprint ?
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
- [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.