A142462 → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(\left(36 - 84\,n + 49\,n^{2}\right)\,P_{n}(x) = \left(-36 - 15600\,x + 2340\,x^{2} - 210\,n + 6580\,n x - 36204\,n x^{2} + 294\,n^{2} + 3528\,n^{2} x + 90797\,n^{2} x^{2} - 1372\,n^{3} x - 78890\,n^{3} x^{2} + 21609\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(2436\,x - 13380\,x^{2} + 3420\,x^{3} + 490\,n x - 3787\,n x^{2} - 44772\,n x^{3} - 686\,n^{2} x + 7448\,n^{2} x^{2} + 97559\,n^{2} x^{3} + 1372\,n^{3} x^{2} - 66542\,n^{3} x^{3} + 9604\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(2352\,x^{2} - 3108\,x^{3} + 756\,x^{4} + 1715\,n x^{2} + 7105\,n x^{3} - 8820\,n x^{4} - 2401\,n^{2} x^{2} - 15092\,n^{2} x^{3} + 17493\,n^{2} x^{4} + 9604\,n^{3} x^{3} - 9604\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(72 - 864\,x + 126\,n + 3024\,n x - 245\,n^{2} - 3528\,n^{2} x + 1372\,n^{3} x\right)\,P_{n-1}(x) + \left(-84\,x - 144\,x^{2} - 833\,n x - 672\,n x^{2} + 686\,n^{2} x + 2156\,n^{2} x^{2} - 1372\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 4 | 5 | ||||||
| 2 | 1 | 64 | 9 | 74 | |||||
| 3 | 1 | 572 | 1287 | 16 | 1876 | ||||
| 4 | 1 | 4664 | 38610 | 18656 | 25 | 61956 | |||
| 5 | 1 | 37428 | 810018 | 1440032 | 233925 | 36 | 2521440 | ||
| 6 | 1 | 299568 | 14592447 | 63361408 | 40534575 | 2696112 | 49 | 121484160 | |
| 7 | 1 | 2396716 | 243151713 | 2146272688 | 3353551075 | 972606852 | 29359771 | 64 | 6747338880 |
Compute more
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Technical details
- Node id
n-5ed35238ff3803e6· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-9b672623d94ceedb
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, 64, 9; 3: 1, 572, 1287, 16; 4: 1, 4664, 38610, 18656, 25; 5: 1, 37428, 810018, 1440032, 233925, 36; 6: 1, 299568, 14592447, 63361408, 40534575, 2696112, 49; 7: 1, 2396716, 243151713, 2146272688, 3353551075, 972606852, 29359771, 64; 8: 1, 19173928, 3879157968, 62779708768, 194505962350, 141254344728, 21119860048, 306782848, 81; 9: 1, 153391652, 60344436420, 1677693668672, 9172031526350, 13207725397944, 5137936860308, 429115992320, 3106180953, 100;
%F FormulaThe row polynomials satisfy (49*n^2-84*n+36)*P(n,x) = (294*n^2-210*n-36-(1372*n^3-3528*n^2-6580*n+15600)*x+(21609*n^4-78890*n^3+90797*n^2-36204*n+2340)*x^2)*P(n-2,x) - ((686*n^2-490*n-2436)*x-(1372*n^3+7448*n^2-3787*n-13380)*x^2-(9604*n^4-66542*n^3+97559*n^2-44772*n+3420)*x^3)*d/dx P(n-2,x) - ((2401*n^2-1715*n-2352)*x^2-(9604*n^3-15092*n^2+7105*n-3108)*x^3+(9604*n^3-17493*n^2+8820*n-756)*x^4)*d^2/dx^2 P(n-2,x) - (245*n^2-126*n-72-(1372*n^3-3528*n^2+3024*n-864)*x)*P(n-1,x) + ((686*n^2-833*n-84)*x-(1372*n^3-2156*n^2+672*n+144)*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[((294*n^2 - 210*n - 36 - (1372*n^3 - 3528*n^2 - 6580*n + 15600)*x + (21609*n^4 - 78890*n^3 + 90797*n^2 - 36204*n + 2340)*x^2)*P[n - 2, x] - ((686*n^2 - 490*n - 2436)*x - (1372*n^3 + 7448*n^2 - 3787*n - 13380)*x^2 - (9604*n^4 - 66542*n^3 + 97559*n^2 - 44772*n + 3420)*x^3)*D[P[n - 2, x], x] - ((2401*n^2 - 1715*n - 2352)*x^2 - (9604*n^3 - 15092*n^2 + 7105*n - 3108)*x^3 + (9604*n^3 - 17493*n^2 + 8820*n - 756)*x^4)*D[P[n - 2, x], {x, 2}] - (245*n^2 - 126*n - 72 - (1372*n^3 - 3528*n^2 + 3024*n - 864)*x)*P[n - 1, x] + ((686*n^2 - 833*n - 84)*x - (1372*n^3 - 2156*n^2 + 672*n + 144)*x^2)*D[P[n - 1, x], x])/(49*n^2 - 84*n + 36)]];
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] = ((294*n^2 - 210*n - 36 - (1372*n^3 - 3528*n^2 - 6580*n + 15600)*x + (21609*n^4 - 78890*n^3 + 90797*n^2 - 36204*n + 2340)*x^2)*v[i - 2] - ((686*n^2 - 490*n - 2436)*x - (1372*n^3 + 7448*n^2 - 3787*n - 13380)*x^2 - (9604*n^4 - 66542*n^3 + 97559*n^2 - 44772*n + 3420)*x^3)*deriv(v[i - 2], x) - ((2401*n^2 - 1715*n - 2352)*x^2 - (9604*n^3 - 15092*n^2 + 7105*n - 3108)*x^3 + (9604*n^3 - 17493*n^2 + 8820*n - 756)*x^4)*deriv(deriv(v[i - 2], x), x) - (245*n^2 - 126*n - 72 - (1372*n^3 - 3528*n^2 + 3024*n - 864)*x)*v[i - 1] + ((686*n^2 - 833*n - 84)*x - (1372*n^3 - 2156*n^2 + 672*n + 144)*x^2)*deriv(v[i - 1], x))/(49*n^2 - 84*n + 36)); 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 ›
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.