A144440 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(49 - 84\,n + 36\,n^{2}\right)\,P_{n}(x) = \left(1701\,x^{2} - 3888\,n x^{2} + 2916\,n^{2} x^{2}\right)\,P_{n-4}(x) + \left(819\,x^{3} - 1728\,n x^{3} + 1296\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(1428\,x - 4095\,x^{2} + 576\,n x + 15282\,n x^{2} - 432\,n^{2} x - 18144\,n^{2} x^{2} + 5832\,n^{3} x^{2}\right)\,P_{n-3}(x) + \left(651\,x^{2} - 4557\,x^{3} - 576\,n x^{2} + 12555\,n x^{3} + 432\,n^{2} x^{2} - 12312\,n^{2} x^{3} + 2592\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(819\,x^{3} - 819\,x^{4} - 1728\,n x^{3} + 1728\,n x^{4} + 1296\,n^{2} x^{3} - 1296\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(14 - 1792\,x - 735\,x^{2} - 96\,n - 540\,n x - 4914\,n x^{2} + 72\,n^{2} + 1728\,n^{2} x + 14013\,n^{2} x^{2} - 432\,n^{3} x - 11340\,n^{3} x^{2} + 2916\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(462\,x - 1904\,x^{2} - 833\,x^{3} + 288\,n x - 576\,n x^{2} - 7098\,n x^{3} - 216\,n^{2} x + 15768\,n^{2} x^{3} + 432\,n^{3} x^{2} - 9288\,n^{3} x^{3} + 1296\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(315\,x^{2} - 168\,x^{3} - 147\,x^{4} + 432\,n x^{2} + 1080\,n x^{3} - 1512\,n x^{4} - 324\,n^{2} x^{2} - 2592\,n^{2} x^{3} + 2916\,n^{2} x^{4} + 1296\,n^{3} x^{3} - 1296\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(35 - 392\,x + 12\,n + 1260\,n x - 36\,n^{2} - 1296\,n^{2} x + 432\,n^{3} x\right)\,P_{n-1}(x) + \left(42\,x - 196\,x^{2} - 396\,n x - 252\,n x^{2} + 216\,n^{2} x + 864\,n^{2} x^{2} - 432\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
\(P_{2}(x) = 1 + 44\,x + 9\,x^{2}\)
\(P_{3}(x) = 1 + 216\,x + 486\,x^{2} + 16\,x^{3}\)
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 | 44 | 9 | 54 | ||||||||||||
| 3 | 1 | 216 | 486 | 16 | 719 | |||||||||||
| 4 | 1 | 916 | 7101 | 3664 | 25 | 11707 | ||||||||||
| 5 | 1 | 3728 | 71775 | 127600 | 23300 | 36 | 226440 | |||||||||
| 6 | 1 | 14988 | 617652 | 2589872 | 1715700 | 134892 | 49 | 5073154 | ||||||||
| 7 | 1 | 60040 | 4888296 | 40556144 | 63368975 | 19553184 | 735490 | 64 | 129162194 | |||||||
| 8 | 1 | 240260 | 36885951 | 547900448 | 1659733375 | 1232776008 | 200823511 | 3844160 | 81 | 3682203795 | ||||||
| 9 | 1 | 961152 | 270499797 | 6751000848 | 35464152000 | 51068378880 | 20674940097 | 1923554112 | 19463328 | 100 | 116172950315 | |||||
| 10 | 1 | 3844732 | 1949185134 | 78286281168 | 664022597025 | 1641358172340 | 1301484290169 | 313145124672 | 17542666206 | 96118300 | 121 | 4017888279868 | ||||
| 11 | 1 | 15379064 | 13893001830 | 870935927664 | 11355029992950 | 44582594148684 | 60681864257931 | 29068876781952 | 4409113133799 | 154366687000 | 465216686 | 144 | 151137154527705 | |||
| 12 | 1 | 61516404 | 98349654465 | 9411317688448 | 182003291525700 | 1075917434174748 | 2312613054338433 | 1912742105199552 | 589690664543268 | 58820735552800 | 1322256465585 | 2214590544 | 169 | 6142621485250117 | ||
| 13 | 1 | 246065776 | 693257394843 | 99607431629200 | 2781870450864625 | 23815851171290712 | 76339443459658974 | 99708660845268864 | 53585665135404102 | 11127481803458500 | 753281201695825 | 11092118317488 | 10396279036 | 196 | 268223657517328142 | |
| 14 | 1 | 984263276 | 4873701975480 | 1038502405870768 | 41033166382444100 | 494004889538904096 | 2265243246501709677 | 4396242744670054080 | 3744585774829356813 | 1372235804274733600 | 198600525291029444 | 9346521652836912 | 91517292650680 | 48228900524 | 225 | 12522427615754729676 |
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Technical details
- Node id
n-7ef7dde6c66d1b62· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-f11fad98cdca98d2
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, 44, 9; 3: 1, 216, 486, 16; 4: 1, 916, 7101, 3664, 25; 5: 1, 3728, 71775, 127600, 23300, 36; 6: 1, 14988, 617652, 2589872, 1715700, 134892, 49; 7: 1, 60040, 4888296, 40556144, 63368975, 19553184, 735490, 64; 8: 1, 240260, 36885951, 547900448, 1659733375, 1232776008, 200823511, 3844160, 81; 9: 1, 961152, 270499797, 6751000848, 35464152000, 51068378880, 20674940097, 1923554112, 19463328, 100;
%F FormulaThe row polynomials satisfy (36*n^2-84*n+49)*P(n,x) = (2916*n^2-3888*n+1701)*x^2*P(n-4,x) + (1296*n^2-1728*n+819)*x^3*d/dx P(n-4,x) - ((432*n^2-576*n-1428)*x-(5832*n^3-18144*n^2+15282*n-4095)*x^2)*P(n-3,x) + ((432*n^2-576*n+651)*x^2+(2592*n^3-12312*n^2+12555*n-4557)*x^3)*d/dx P(n-3,x) + ((1296*n^2-1728*n+819)*x^3-(1296*n^2-1728*n+819)*x^4)*d^2/dx^2 P(n-3,x) + (72*n^2-96*n+14-(432*n^3-1728*n^2+540*n+1792)*x+(2916*n^4-11340*n^3+14013*n^2-4914*n-735)*x^2)*P(n-2,x) - ((216*n^2-288*n-462)*x-(432*n^3-576*n-1904)*x^2-(1296*n^4-9288*n^3+15768*n^2-7098*n-833)*x^3)*d/dx P(n-2,x) - ((324*n^2-432*n-315)*x^2-(1296*n^3-2592*n^2+1080*n-168)*x^3+(1296*n^3-2916*n^2+1512*n+147)*x^4)*d^2/dx^2 P(n-2,x) - (36*n^2-12*n-35-(432*n^3-1296*n^2+1260*n-392)*x)*P(n-1,x) + ((216*n^2-396*n+42)*x-(432*n^3-864*n^2+252*n+196)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 44*x + 9*x^2, P(3,x) = 1 + 216*x + 486*x^2 + 16*x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 1 + 44*x + 9*x^2;
P[3, x_] := 1 + 216*x + 486*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((2916*n^2 - 3888*n + 1701)*x^2*P[n - 4, x] + (1296*n^2 - 1728*n + 819)*x^3*D[P[n - 4, x], x] - ((432*n^2 - 576*n - 1428)*x - (5832*n^3 - 18144*n^2 + 15282*n - 4095)*x^2)*P[n - 3, x] + ((432*n^2 - 576*n + 651)*x^2 + (2592*n^3 - 12312*n^2 + 12555*n - 4557)*x^3)*D[P[n - 3, x], x] + ((1296*n^2 - 1728*n + 819)*x^3 - (1296*n^2 - 1728*n + 819)*x^4)*D[P[n - 3, x], {x, 2}] + (72*n^2 - 96*n + 14 - (432*n^3 - 1728*n^2 + 540*n + 1792)*x + (2916*n^4 - 11340*n^3 + 14013*n^2 - 4914*n - 735)*x^2)*P[n - 2, x] - ((216*n^2 - 288*n - 462)*x - (432*n^3 - 576*n - 1904)*x^2 - (1296*n^4 - 9288*n^3 + 15768*n^2 - 7098*n - 833)*x^3)*D[P[n - 2, x], x] - ((324*n^2 - 432*n - 315)*x^2 - (1296*n^3 - 2592*n^2 + 1080*n - 168)*x^3 + (1296*n^3 - 2916*n^2 + 1512*n + 147)*x^4)*D[P[n - 2, x], {x, 2}] - (36*n^2 - 12*n - 35 - (432*n^3 - 1296*n^2 + 1260*n - 392)*x)*P[n - 1, x] + ((216*n^2 - 396*n + 42)*x - (432*n^3 - 864*n^2 + 252*n + 196)*x^2)*D[P[n - 1, x], x])/(36*n^2 - 84*n + 49)]];
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; v[3] = 1 + 44*x + 9*x^2; v[4] = 1 + 216*x + 486*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((2916*n^2 - 3888*n + 1701)*x^2*v[i - 4] + (1296*n^2 - 1728*n + 819)*x^3*deriv(v[i - 4], x) - ((432*n^2 - 576*n - 1428)*x - (5832*n^3 - 18144*n^2 + 15282*n - 4095)*x^2)*v[i - 3] + ((432*n^2 - 576*n + 651)*x^2 + (2592*n^3 - 12312*n^2 + 12555*n - 4557)*x^3)*deriv(v[i - 3], x) + ((1296*n^2 - 1728*n + 819)*x^3 - (1296*n^2 - 1728*n + 819)*x^4)*deriv(deriv(v[i - 3], x), x) + (72*n^2 - 96*n + 14 - (432*n^3 - 1728*n^2 + 540*n + 1792)*x + (2916*n^4 - 11340*n^3 + 14013*n^2 - 4914*n - 735)*x^2)*v[i - 2] - ((216*n^2 - 288*n - 462)*x - (432*n^3 - 576*n - 1904)*x^2 - (1296*n^4 - 9288*n^3 + 15768*n^2 - 7098*n - 833)*x^3)*deriv(v[i - 2], x) - ((324*n^2 - 432*n - 315)*x^2 - (1296*n^3 - 2592*n^2 + 1080*n - 168)*x^3 + (1296*n^3 - 2916*n^2 + 1512*n + 147)*x^4)*deriv(deriv(v[i - 2], x), x) - (36*n^2 - 12*n - 35 - (432*n^3 - 1296*n^2 + 1260*n - 392)*x)*v[i - 1] + ((216*n^2 - 396*n + 42)*x - (432*n^3 - 864*n^2 + 252*n + 196)*x^2)*deriv(v[i - 1], x))/(36*n^2 - 84*n + 49)); 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
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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 T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = …?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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from
Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = …?
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.