A290316 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(169 - 156\,n + 36\,n^{2}\right)\,P_{n}(x) = \left(182 + 10560\,x + 1679184\,x^{2} - 240\,n - 27000\,n x - 3493854\,n x^{2} + 72\,n^{2} + 15552\,n^{2} x + 2644812\,n^{2} x^{2} - 2592\,n^{3} x - 868968\,n^{3} x^{2} + 104976\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-402\,x - 8094\,x^{2} + 1210716\,x^{3} + 720\,n x + 13284\,n x^{2} - 2340468\,n x^{3} - 216\,n^{2} x - 10368\,n^{2} x^{2} + 1605744\,n^{2} x^{3} + 2592\,n^{3} x^{2} - 462672\,n^{3} x^{3} + 46656\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-711\,x^{2} - 45018\,x^{3} + 141453\,x^{4} + 1080\,n x^{2} + 76788\,n x^{3} - 240084\,n x^{4} - 324\,n^{2} x^{2} - 42768\,n^{2} x^{3} + 131220\,n^{2} x^{4} + 7776\,n^{3} x^{3} - 23328\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-13 - 20280\,x + 84\,n + 30888\,n x - 36\,n^{2} - 15552\,n^{2} x + 2592\,n^{3} x\right)\,P_{n-1}(x) + \left(744\,x + 14196\,x^{2} - 828\,n x - 25272\,n x^{2} + 216\,n^{2} x + 14256\,n^{2} x^{2} - 2592\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 24\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | 1 | 24 | 25 | |||||||||||||
| 3 | 1 | 192 | 810 | 1003 | ||||||||||||
| 4 | 1 | 936 | 19926 | 34560 | 55423 | |||||||||||
| 5 | 1 | 3984 | 246888 | 1833408 | 1782000 | 3866281 | ||||||||||
| 6 | 1 | 16248 | 2346732 | 44668800 | 169265700 | 107775360 | 324072841 | |||||||||
| 7 | 1 | 65376 | 19607670 | 772101504 | 7016309100 | 16348962240 | 7481406240 | 31638452131 | ||||||||
| 8 | 1 | 261960 | 152698770 | 11067390720 | 196169566500 | 1049655233952 | 1677586755600 | 586297958400 | 3520929865903 | |||||||
| 9 | 1 | 1048368 | 1141388820 | 141620857920 | 4365948154500 | 43912964616192 | 156170822310216 | 183806985369600 | 51200301398400 | 439599785144017 | ||||||
| 10 | 1 | 4194072 | 8322316488 | 1684054488960 | 83975084640000 | 1419321482901792 | 9243140150291544 | 23625662500048896 | 21518879814640800 | 4930399393920000 | 60823070807442553 | |||||
| 11 | 1 | 16776960 | 59756644674 | 19059829286400 | 1462694354919000 | 38671127814799872 | 413237734451347680 | 1896049279828159488 | 3677601051036580320 | 2688436938839040000 | 519023144197958400 | 9234501090125512795 | ||||
| 12 | 1 | 67108584 | 424977937038 | 208445610739968 | 23750202952287000 | 934680025699301952 | 15273150901032614688 | 112958642944892676096 | 387101572759163773296 | 592864530359308128000 | 357655630549636742400 | 59297321862748569600 | 1526109488476089878623 | |||
| 13 | 1 | 268435152 | 3004289450496 | 2224982145246912 | 366400877600003400 | 20700529509406102272 | 492470202681376943184 | 5477380933561375629312 | 29835362148254395171344 | 79683426114516946195200 | 99323352600414567704640 | 50539041431786647756800 | 7307159558711620608000 | 272679262148300639246713 | ||
| 14 | 1 | 1073741496 | 21158673700068 | 23339377093182336 | 5441135656179962700 | 429345283206882547008 | 14344036418587248726192 | 228564413583863833092096 | 1848121382508499828188432 | 7756202995698029154825600 | 16673813260010587789206912 | 17321123162636779932241920 | 7565886113690031677222400 | 966101616448831070208000 | 52374591790774610436845161 | |
| 15 | 1 | 4294966944 | 148668793043022 | 241744394152116864 | 78502399969164261300 | 8467912541669546594880 | 386492085870984468482688 | 8513139742250363692038144 | 97405034524460242189732368 | 600487748399826262181606400 | 2009104105402672050562311456 | 3565400554605920734612500480 | 3146222812890363317928011520 | 1196896168405814186909491200 | 136412562424598978918400000 | 10760837165138611827413557267 |
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Technical details
- Node id
n-f94ce942a2de57b8· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-60bf375b3a8b0a4e
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, 24; 3: 1, 192, 810; 4: 1, 936, 19926, 34560; 5: 1, 3984, 246888, 1833408, 1782000; 6: 1, 16248, 2346732, 44668800, 169265700, 107775360; 7: 1, 65376, 19607670, 772101504, 7016309100, 16348962240, 7481406240; 8: 1, 261960, 152698770, 11067390720, 196169566500, 1049655233952, 1677586755600, 586297958400; 9: 1, 1048368, 1141388820, 141620857920, 4365948154500, 43912964616192, 156170822310216, 183806985369600, 51200301398400; 10: 1, 4194072, 8322316488, 1684054488960, 83975084640000, 1419321482901792, 9243140150291544, 23625662500048896, 21518879814640800, 4930399393920000;
%F FormulaThe row polynomials satisfy (36*n^2-156*n+169)*P(n,x) = (72*n^2-240*n+182-(2592*n^3-15552*n^2+27000*n-10560)*x+(104976*n^4-868968*n^3+2644812*n^2-3493854*n+1679184)*x^2)*P(n-2,x) - ((216*n^2-720*n+402)*x-(2592*n^3-10368*n^2+13284*n-8094)*x^2-(46656*n^4-462672*n^3+1605744*n^2-2340468*n+1210716)*x^3)*d/dx P(n-2,x) - ((324*n^2-1080*n+711)*x^2-(7776*n^3-42768*n^2+76788*n-45018)*x^3+(23328*n^3-131220*n^2+240084*n-141453)*x^4)*d^2/dx^2 P(n-2,x) - (36*n^2-84*n+13-(2592*n^3-15552*n^2+30888*n-20280)*x)*P(n-1,x) + ((216*n^2-828*n+744)*x-(2592*n^3-14256*n^2+25272*n-14196)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 24*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 1 + 24*x;
P[n_, x_] := P[n, x] = Expand[Together[((72*n^2 - 240*n + 182 - (2592*n^3 - 15552*n^2 + 27000*n - 10560)*x + (104976*n^4 - 868968*n^3 + 2644812*n^2 - 3493854*n + 1679184)*x^2)*P[n - 2, x] - ((216*n^2 - 720*n + 402)*x - (2592*n^3 - 10368*n^2 + 13284*n - 8094)*x^2 - (46656*n^4 - 462672*n^3 + 1605744*n^2 - 2340468*n + 1210716)*x^3)*D[P[n - 2, x], x] - ((324*n^2 - 1080*n + 711)*x^2 - (7776*n^3 - 42768*n^2 + 76788*n - 45018)*x^3 + (23328*n^3 - 131220*n^2 + 240084*n - 141453)*x^4)*D[P[n - 2, x], {x, 2}] - (36*n^2 - 84*n + 13 - (2592*n^3 - 15552*n^2 + 30888*n - 20280)*x)*P[n - 1, x] + ((216*n^2 - 828*n + 744)*x - (2592*n^3 - 14256*n^2 + 25272*n - 14196)*x^2)*D[P[n - 1, x], x])/(36*n^2 - 156*n + 169)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 24*x; for(n = 3, N, my(i = n); v[i] = ((72*n^2 - 240*n + 182 - (2592*n^3 - 15552*n^2 + 27000*n - 10560)*x + (104976*n^4 - 868968*n^3 + 2644812*n^2 - 3493854*n + 1679184)*x^2)*v[i - 2] - ((216*n^2 - 720*n + 402)*x - (2592*n^3 - 10368*n^2 + 13284*n - 8094)*x^2 - (46656*n^4 - 462672*n^3 + 1605744*n^2 - 2340468*n + 1210716)*x^3)*deriv(v[i - 2], x) - ((324*n^2 - 1080*n + 711)*x^2 - (7776*n^3 - 42768*n^2 + 76788*n - 45018)*x^3 + (23328*n^3 - 131220*n^2 + 240084*n - 141453)*x^4)*deriv(deriv(v[i - 2], x), x) - (36*n^2 - 84*n + 13 - (2592*n^3 - 15552*n^2 + 30888*n - 20280)*x)*v[i - 1] + ((216*n^2 - 828*n + 744)*x - (2592*n^3 - 14256*n^2 + 25272*n - 14196)*x^2)*deriv(v[i - 1], x))/(36*n^2 - 156*n + 169)); 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]
-
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
-
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
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.