A257608 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(64 - 144\,n + 81\,n^{2}\right)\,P_{n}(x) = \left(-64 - 44608\,x + 9792\,x^{2} - 504\,n + 19332\,n x - 115344\,n x^{2} + 648\,n^{2} + 7776\,n^{2} x + 269001\,n^{2} x^{2} - 2916\,n^{3} x - 223074\,n^{3} x^{2} + 59049\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(6624\,x - 37328\,x^{2} + 13504\,x^{3} + 1134\,n x - 12123\,n x^{2} - 139824\,n x^{3} - 1458\,n^{2} x + 21384\,n^{2} x^{2} + 284067\,n^{2} x^{3} + 2916\,n^{3} x^{2} - 185166\,n^{3} x^{3} + 26244\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(6480\,x^{2} - 9360\,x^{3} + 2880\,x^{4} + 5103\,n x^{2} + 22113\,n x^{3} - 27216\,n x^{4} - 6561\,n^{2} x^{2} - 43740\,n^{2} x^{3} + 50301\,n^{2} x^{4} + 26244\,n^{3} x^{3} - 26244\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(128 - 2048\,x + 360\,n + 6912\,n x - 567\,n^{2} - 7776\,n^{2} x + 2916\,n^{3} x\right)\,P_{n-1}(x) + \left(-144\,x - 256\,x^{2} - 1863\,n x - 1728\,n x^{2} + 1458\,n^{2} x + 4860\,n^{2} x^{2} - 2916\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 1 | 4 | 5 | |||||||||||||
| 2 | 1 | 80 | 9 | 90 | ||||||||||||
| 3 | 1 | 876 | 1971 | 16 | 2864 | |||||||||||
| 4 | 1 | 8872 | 74898 | 35488 | 25 | 119284 | ||||||||||
| 5 | 1 | 88868 | 1981998 | 3523552 | 555425 | 36 | 6149880 | |||||||||
| 6 | 1 | 888864 | 45056223 | 197318912 | 125156175 | 7999776 | 49 | 376420000 | ||||||||
| 7 | 1 | 8888860 | 948065661 | 8488627760 | 13263480875 | 3792262644 | 108888535 | 64 | 26610214400 | |||||||
| 8 | 1 | 88888856 | 19113243984 | 315212280224 | 981497584750 | 709227630504 | 104060995024 | 1422221696 | 81 | 2130622845120 | ||||||
| 9 | 1 | 888888852 | 375951630960 | 10694794369152 | 58971293276850 | 84918662318664 | 32752807755528 | 2673433820160 | 17999999253 | 100 | 190405832059520 | |||||
| 10 | 1 | 8888888848 | 7289080982181 | 342229183458816 | 3101021742342450 | 7812516933317088 | 6078002614991202 | 1368916733835264 | 65601728839629 | 222222221200 | 121 | 18775809128876800 | ||||
| 11 | 1 | 88888888844 | 140132538653895 | 10528377868756560 | 148960722812552250 | 604976700926108088 | 823440509593869342 | 381339450400133760 | 53299912960580085 | 1557028207265500 | 2688888887531 | 144 | 2024245613085696000 | |||
| 12 | 1 | 888888888840 | 2680718234414814 | 315222790404507040 | 6712439844719478375 | 41557148457005787408 | 90578456055325627620 | 73879375034676955392 | 21748305096891109935 | 1970142440028169000 | 36040767373799166 | 31999999998240 | 169 | 236799844093548736000 | ||
| 13 | 1 | 8888888888836 | 51133646453870466 | 9259918770138193696 | 288748194275198164375 | 2617240505499177787548 | 8601904459742080333308 | 11235140518846798802688 | 5888791137373150021983 | 1154992777100792657500 | 70028135699170089826 | 818138343261927456 | 375555555553321 | 196 | 29866975303740666291200 | |
| 14 | 1 | 88888888888832 | 973719282623525883 | 268368151600113061888 | 12000327888311356498025 | 154488450032977577476608 | 733156669957218833500251 | 1438097986412390246744064 | 1211952862582341337010619 | 429134583424937715212800 | 58081586979426965450441 | 2415313364401017556992 | 18284284307041763803 | 4355555555552768 | 225 | 4039615411241639272243200 |
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Technical details
- Node id
n-f3b058123135c08b· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-93317769465d2bb7
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: 1, 4; 2: 1, 80, 9; 3: 1, 876, 1971, 16; 4: 1, 8872, 74898, 35488, 25; 5: 1, 88868, 1981998, 3523552, 555425, 36; 6: 1, 888864, 45056223, 197318912, 125156175, 7999776, 49; 7: 1, 8888860, 948065661, 8488627760, 13263480875, 3792262644, 108888535, 64; 8: 1, 88888856, 19113243984, 315212280224, 981497584750, 709227630504, 104060995024, 1422221696, 81; 9: 1, 888888852, 375951630960, 10694794369152, 58971293276850, 84918662318664, 32752807755528, 2673433820160, 17999999253, 100;
%F FormulaThe row polynomials satisfy (81*n^2-144*n+64)*P(n,x) = (648*n^2-504*n-64-(2916*n^3-7776*n^2-19332*n+44608)*x+(59049*n^4-223074*n^3+269001*n^2-115344*n+9792)*x^2)*P(n-2,x) - ((1458*n^2-1134*n-6624)*x-(2916*n^3+21384*n^2-12123*n-37328)*x^2-(26244*n^4-185166*n^3+284067*n^2-139824*n+13504)*x^3)*d/dx P(n-2,x) - ((6561*n^2-5103*n-6480)*x^2-(26244*n^3-43740*n^2+22113*n-9360)*x^3+(26244*n^3-50301*n^2+27216*n-2880)*x^4)*d^2/dx^2 P(n-2,x) - (567*n^2-360*n-128-(2916*n^3-7776*n^2+6912*n-2048)*x)*P(n-1,x) + ((1458*n^2-1863*n-144)*x-(2916*n^3-4860*n^2+1728*n+256)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((648*n^2 - 504*n - 64 - (2916*n^3 - 7776*n^2 - 19332*n + 44608)*x + (59049*n^4 - 223074*n^3 + 269001*n^2 - 115344*n + 9792)*x^2)*P[n - 2, x] - ((1458*n^2 - 1134*n - 6624)*x - (2916*n^3 + 21384*n^2 - 12123*n - 37328)*x^2 - (26244*n^4 - 185166*n^3 + 284067*n^2 - 139824*n + 13504)*x^3)*D[P[n - 2, x], x] - ((6561*n^2 - 5103*n - 6480)*x^2 - (26244*n^3 - 43740*n^2 + 22113*n - 9360)*x^3 + (26244*n^3 - 50301*n^2 + 27216*n - 2880)*x^4)*D[P[n - 2, x], {x, 2}] - (567*n^2 - 360*n - 128 - (2916*n^3 - 7776*n^2 + 6912*n - 2048)*x)*P[n - 1, x] + ((1458*n^2 - 1863*n - 144)*x - (2916*n^3 - 4860*n^2 + 1728*n + 256)*x^2)*D[P[n - 1, x], x])/(81*n^2 - 144*n + 64)]];
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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((648*n^2 - 504*n - 64 - (2916*n^3 - 7776*n^2 - 19332*n + 44608)*x + (59049*n^4 - 223074*n^3 + 269001*n^2 - 115344*n + 9792)*x^2)*v[i - 2] - ((1458*n^2 - 1134*n - 6624)*x - (2916*n^3 + 21384*n^2 - 12123*n - 37328)*x^2 - (26244*n^4 - 185166*n^3 + 284067*n^2 - 139824*n + 13504)*x^3)*deriv(v[i - 2], x) - ((6561*n^2 - 5103*n - 6480)*x^2 - (26244*n^3 - 43740*n^2 + 22113*n - 9360)*x^3 + (26244*n^3 - 50301*n^2 + 27216*n - 2880)*x^4)*deriv(deriv(v[i - 2], x), x) - (567*n^2 - 360*n - 128 - (2916*n^3 - 7776*n^2 + 6912*n - 2048)*x)*v[i - 1] + ((1458*n^2 - 1863*n - 144)*x - (2916*n^3 - 4860*n^2 + 1728*n + 256)*x^2)*deriv(v[i - 1], x))/(81*n^2 - 144*n + 64)); 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]
-
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
-
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 ≥ 1
Proof chains ? JSON
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.