A257617 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(25 - 70\,n + 49\,n^{2}\right)\,P_{n}(x) = \left(-100 - 12720\,x - 1200\,x^{2} - 210\,n + 5404\,n x - 14840\,n x^{2} + 490\,n^{2} + 5880\,n^{2} x + 59633\,n^{2} x^{2} - 2744\,n^{3} x - 66542\,n^{3} x^{2} + 21609\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(2485\,x - 11850\,x^{2} - 1175\,x^{3} + 588\,n x - 1435\,n x^{2} - 20860\,n x^{3} - 1372\,n^{2} x + 6468\,n^{2} x^{2} + 70217\,n^{2} x^{3} + 2744\,n^{3} x^{2} - 61054\,n^{3} x^{3} + 9604\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(2205\,x^{2} - 2030\,x^{3} - 175\,x^{4} + 1029\,n x^{2} + 3381\,n x^{3} - 4410\,n x^{4} - 2401\,n^{2} x^{2} - 10976\,n^{2} x^{3} + 13377\,n^{2} x^{4} + 9604\,n^{3} x^{3} - 9604\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(100 - 500\,x - 35\,n + 2100\,n x - 147\,n^{2} - 2940\,n^{2} x + 1372\,n^{3} x\right)\,P_{n-1}(x) + \left(-140\,x - 200\,x^{2} - 637\,n x - 140\,n x^{2} + 686\,n^{2} x + 1568\,n^{2} x^{2} - 1372\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 2 | 8 | 10 | |||||||||||||
| 2 | 4 | 144 | 36 | 184 | ||||||||||||
| 3 | 8 | 1552 | 3492 | 128 | 5180 | |||||||||||
| 4 | 16 | 14704 | 111744 | 58816 | 400 | 185680 | ||||||||||
| 5 | 32 | 134256 | 2548836 | 4531264 | 839100 | 1152 | 8054640 | |||||||||
| 6 | 64 | 1213040 | 49843656 | 208438144 | 138454600 | 10917360 | 3136 | 408870000 | ||||||||
| 7 | 128 | 10928624 | 898484076 | 7452405632 | 11644383800 | 3593936304 | 133875644 | 8192 | 23734022400 | |||||||
| 8 | 256 | 98383728 | 15457678992 | 230505615424 | 698663028000 | 518637634704 | 84158474512 | 1574139648 | 20736 | 1549094976000 | ||||||
| 9 | 512 | 885512944 | 258612396660 | 6510763155904 | 34285996731700 | 49371835293648 | 19939212164956 | 1839021487360 | 17931637116 | 51200 | 112224258432000 | |||||
| 10 | 1024 | 7969749616 | 4253357785752 | 173195076549632 | 1476194868919400 | 3653515811729952 | 2893341943082024 | 692780306198528 | 38280220071768 | 199243740400 | 123904 | 8931768797952000 | ||||
| 11 | 2048 | 71728041456 | 69219300453372 | 4422055207883520 | 58087328730130800 | 228711791928741408 | 311302161236342472 | 148703561549134848 | 22386654489910320 | 769103338370800 | 2169773254044 | 294912 | 774454117282560000 | |||
| 12 | 4096 | 645553020272 | 1119128749969824 | 109705944500377280 | 2143368615118368000 | 12728269723304238048 | 27394590188798137536 | 22628035063651978752 | 6944514312983512320 | 685662153127358000 | 15046064305149856 | 23239908729792 | 692224 | 72650335080001536000 | ||
| 13 | 8192 | 5809978591472 | 18020807048870532 | 2666485203504814912 | 75443068441870607500 | 649960126496942903136 | 2088916024933153909416 | 2728380114198405106176 | 1462410284618121532056 | 301772273767482430000 | 20165294351505162772 | 288332912781928512 | 245471595489692 | 1605632 | 7330020277012392960000 | |
| 14 | 16384 | 52289810370672 | 289457143639378344 | 63860081915029893248 | 2563271638650409902600 | 31109995886545976278032 | 143223046194400764534096 | 278294771648237320829952 | 236756464117274733209424 | 86416655240405489661200 | 12406234731067983928584 | 574740737235269039232 | 5435361919450548904 | 2562200708162928 | 3686400 | 791414767709286589440000 |
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Technical details
- Node id
n-c6b14968e3b3956f· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-b45811e67d078769
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, 8; 2: 4, 144, 36; 3: 8, 1552, 3492, 128; 4: 16, 14704, 111744, 58816, 400; 5: 32, 134256, 2548836, 4531264, 839100, 1152; 6: 64, 1213040, 49843656, 208438144, 138454600, 10917360, 3136; 7: 128, 10928624, 898484076, 7452405632, 11644383800, 3593936304, 133875644, 8192; 8: 256, 98383728, 15457678992, 230505615424, 698663028000, 518637634704, 84158474512, 1574139648, 20736; 9: 512, 885512944, 258612396660, 6510763155904, 34285996731700, 49371835293648, 19939212164956, 1839021487360, 17931637116, 51200;
%F FormulaThe row polynomials satisfy (49*n^2-70*n+25)*P(n,x) = (490*n^2-210*n-100-(2744*n^3-5880*n^2-5404*n+12720)*x+(21609*n^4-66542*n^3+59633*n^2-14840*n-1200)*x^2)*P(n-2,x) - ((1372*n^2-588*n-2485)*x-(2744*n^3+6468*n^2-1435*n-11850)*x^2-(9604*n^4-61054*n^3+70217*n^2-20860*n-1175)*x^3)*d/dx P(n-2,x) - ((2401*n^2-1029*n-2205)*x^2-(9604*n^3-10976*n^2+3381*n-2030)*x^3+(9604*n^3-13377*n^2+4410*n+175)*x^4)*d^2/dx^2 P(n-2,x) - (147*n^2+35*n-100-(1372*n^3-2940*n^2+2100*n-500)*x)*P(n-1,x) + ((686*n^2-637*n-140)*x-(1372*n^3-1568*n^2+140*n+200)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((490*n^2 - 210*n - 100 - (2744*n^3 - 5880*n^2 - 5404*n + 12720)*x + (21609*n^4 - 66542*n^3 + 59633*n^2 - 14840*n - 1200)*x^2)*P[n - 2, x] - ((1372*n^2 - 588*n - 2485)*x - (2744*n^3 + 6468*n^2 - 1435*n - 11850)*x^2 - (9604*n^4 - 61054*n^3 + 70217*n^2 - 20860*n - 1175)*x^3)*D[P[n - 2, x], x] - ((2401*n^2 - 1029*n - 2205)*x^2 - (9604*n^3 - 10976*n^2 + 3381*n - 2030)*x^3 + (9604*n^3 - 13377*n^2 + 4410*n + 175)*x^4)*D[P[n - 2, x], {x, 2}] - (147*n^2 + 35*n - 100 - (1372*n^3 - 2940*n^2 + 2100*n - 500)*x)*P[n - 1, x] + ((686*n^2 - 637*n - 140)*x - (1372*n^3 - 1568*n^2 + 140*n + 200)*x^2)*D[P[n - 1, x], x])/(49*n^2 - 70*n + 25)]];
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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((490*n^2 - 210*n - 100 - (2744*n^3 - 5880*n^2 - 5404*n + 12720)*x + (21609*n^4 - 66542*n^3 + 59633*n^2 - 14840*n - 1200)*x^2)*v[i - 2] - ((1372*n^2 - 588*n - 2485)*x - (2744*n^3 + 6468*n^2 - 1435*n - 11850)*x^2 - (9604*n^4 - 61054*n^3 + 70217*n^2 - 20860*n - 1175)*x^3)*deriv(v[i - 2], x) - ((2401*n^2 - 1029*n - 2205)*x^2 - (9604*n^3 - 10976*n^2 + 3381*n - 2030)*x^3 + (9604*n^3 - 13377*n^2 + 4410*n + 175)*x^4)*deriv(deriv(v[i - 2], x), x) - (147*n^2 + 35*n - 100 - (1372*n^3 - 2940*n^2 + 2100*n - 500)*x)*v[i - 1] + ((686*n^2 - 637*n - 140)*x - (1372*n^3 - 1568*n^2 + 140*n + 200)*x^2)*deriv(v[i - 1], x))/(49*n^2 - 70*n + 25)); 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.