A104029 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(24\right)\,P_{n}(x) = \left(-152 - 876\,x + 540\,x^{2} + 368\,x^{3} - 24\,x^{4}\right)\,P_{n-4}(x) + \left(-27 - 119\,x + 45\,x^{2} + 432\,x^{3} + 128\,x^{4}\right)\,P_{n-4}'(x) + \left(-27\,x + 9\,x^{2} + 9\,x^{3} + 36\,x^{4}\right)\,P_{n-4}''(x) + \left(568 + 1400\,x - 1344\,x^{2} + 96\,x^{3}\right)\,P_{n-3}(x) + \left(92 + 396\,x - 650\,x^{2} - 1056\,x^{3}\right)\,P_{n-3}'(x) + \left(92\,x - 76\,x^{2} - 130\,x^{3} - 96\,x^{4}\right)\,P_{n-3}''(x) + \left(-596 - 336\,x - 144\,x^{2}\right)\,P_{n-2}(x) + \left(-103 - 300\,x + 960\,x^{2}\right)\,P_{n-2}'(x) + \left(-103\,x + 152\,x^{2} + 192\,x^{3}\right)\,P_{n-2}''(x) + \left(160 + 96\,x\right)\,P_{n-1}(x) + \left(38 - 32\,x\right)\,P_{n-1}'(x) + \left(38\,x - 96\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)
\(P_{2}(x) = 3 + 20\,x + 9\,x^{2}\)
\(P_{3}(x) = 4 + 52\,x + 81\,x^{2} + 16\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 4 | 6 | ||||||
| 2 | 3 | 20 | 9 | 32 | |||||
| 3 | 4 | 52 | 81 | 16 | 153 | ||||
| 4 | 5 | 104 | 315 | 224 | 25 | 673 | |||
| 5 | 6 | 180 | 864 | 1200 | 500 | 36 | 2786 | ||
| 6 | 7 | 284 | 1944 | 4272 | 3500 | 972 | 49 | 11028 | |
| 7 | 8 | 420 | 3843 | 12000 | 15575 | 8568 | 1715 | 64 | 42193 |
Compute more
?
Technical details
- Node id
n-e23e53f7c5877e4e· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-325fa020c74e3c7c
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 0)
- 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: 2, 4; 2: 3, 20, 9; 3: 4, 52, 81, 16; 4: 5, 104, 315, 224, 25; 5: 6, 180, 864, 1200, 500, 36; 6: 7, 284, 1944, 4272, 3500, 972, 49; 7: 8, 420, 3843, 12000, 15575, 8568, 1715, 64; 8: 9, 592, 6930, 28800, 53075, 46368, 18522, 2816, 81; 9: 10, 804, 11664, 61728, 151150, 187596, 119364, 36480, 4374, 100;
%F FormulaThe row polynomials satisfy 24*P(n,x) = -(152+876*x-540*x^2-368*x^3+24*x^4)*P(n-4,x) - (27+119*x-45*x^2-432*x^3-128*x^4)*d/dx P(n-4,x) - (27*x-9*x^2-9*x^3-36*x^4)*d^2/dx^2 P(n-4,x) + (568+1400*x-1344*x^2+96*x^3)*P(n-3,x) + (92+396*x-650*x^2-1056*x^3)*d/dx P(n-3,x) + (92*x-76*x^2-130*x^3-96*x^4)*d^2/dx^2 P(n-3,x) - (596+336*x+144*x^2)*P(n-2,x) - (103+300*x-960*x^2)*d/dx P(n-2,x) - (103*x-152*x^2-192*x^3)*d^2/dx^2 P(n-2,x) + (160+96*x)*P(n-1,x) + (38-32*x)*d/dx P(n-1,x) + (38*x-96*x^2)*d^2/dx^2 P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 3 + 20*x + 9*x^2, P(3,x) = 4 + 52*x + 81*x^2 + 16*x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 3 + 20*x + 9*x^2;
P[3, x_] := 4 + 52*x + 81*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(152 + 876*x - 540*x^2 - 368*x^3 + 24*x^4)*P[n - 4, x] - (27 + 119*x - 45*x^2 - 432*x^3 - 128*x^4)*D[P[n - 4, x], x] - (27*x - 9*x^2 - 9*x^3 - 36*x^4)*D[P[n - 4, x], {x, 2}] + (568 + 1400*x - 1344*x^2 + 96*x^3)*P[n - 3, x] + (92 + 396*x - 650*x^2 - 1056*x^3)*D[P[n - 3, x], x] + (92*x - 76*x^2 - 130*x^3 - 96*x^4)*D[P[n - 3, x], {x, 2}] - (596 + 336*x + 144*x^2)*P[n - 2, x] - (103 + 300*x - 960*x^2)*D[P[n - 2, x], x] - (103*x - 152*x^2 - 192*x^3)*D[P[n - 2, x], {x, 2}] + (160 + 96*x)*P[n - 1, x] + (38 - 32*x)*D[P[n - 1, x], x] + (38*x - 96*x^2)*D[P[n - 1, x], {x, 2}])/(24)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; v[3] = 3 + 20*x + 9*x^2; v[4] = 4 + 52*x + 81*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(152 + 876*x - 540*x^2 - 368*x^3 + 24*x^4)*v[i - 4] - (27 + 119*x - 45*x^2 - 432*x^3 - 128*x^4)*deriv(v[i - 4], x) - (27*x - 9*x^2 - 9*x^3 - 36*x^4)*deriv(deriv(v[i - 4], x), x) + (568 + 1400*x - 1344*x^2 + 96*x^3)*v[i - 3] + (92 + 396*x - 650*x^2 - 1056*x^3)*deriv(v[i - 3], x) + (92*x - 76*x^2 - 130*x^3 - 96*x^4)*deriv(deriv(v[i - 3], x), x) - (596 + 336*x + 144*x^2)*v[i - 2] - (103 + 300*x - 960*x^2)*deriv(v[i - 2], x) - (103*x - 152*x^2 - 192*x^3)*deriv(deriv(v[i - 2], x), x) + (160 + 96*x)*v[i - 1] + (38 - 32*x)*deriv(v[i - 1], x) + (38*x - 96*x^2)*deriv(deriv(v[i - 1], x), x))/(24)); 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
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, of pairwise sums of trinomial coefficients (A027907).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).?
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.