A108426 → D+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-4\,n - 4\,n x + 2\,n x^{2} - 8\,n^{2} - 2\,n^{2} x + 4\,n^{2} x^{2} + 4\,n^{3} + 12\,n^{3} x + 8\,n^{4}\right)\,P_{n}(x) = \left(-24 - 30\,x - 2\,x^{2} + 5\,x^{3} + x^{4} + 120\,n + 153\,n x + 17\,n x^{2} - 20\,n x^{3} - 2\,n x^{4} - 150\,n^{2} - 232\,n^{2} x - 67\,n^{2} x^{2} + 18\,n^{2} x^{3} + n^{2} x^{4} + 73\,n^{3} x + 68\,n^{3} x^{2} + 3\,n^{3} x^{3} + 54\,n^{4} + 48\,n^{4} x + 2\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-8\,x - 10\,x^{2} + 2\,x^{3} + 5\,x^{4} + x^{5} + 32\,n x + 45\,n x^{2} + 6\,n x^{3} - 8\,n x^{4} - n x^{5} - 18\,n^{2} x - 45\,n^{2} x^{2} - 30\,n^{2} x^{3} - 3\,n^{2} x^{4} - 18\,n^{3} x - 20\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 1 | 3 | ||||||
| 2 | 8 | 9 | 2 | 19 | |||||
| 3 | 40 | 70 | 36 | 5 | 151 | ||||
| 4 | 220 | 525 | 432 | 140 | 14 | 1331 | |||
| 5 | 1274 | 3861 | 4400 | 2310 | 540 | 42 | 12427 | ||
| 6 | 7616 | 28028 | 40950 | 30030 | 11440 | 2079 | 132 | 120275 | |
| 7 | 46512 | 201552 | 359856 | 340340 | 182000 | 54054 | 8008 | 429 | 1192751 |
Compute more
?
Technical details
- Node id
n-b5fb99ffd0b8f8fd· revision 0 · JSON- Definition source
derived:one_plus_derivative- Fingerprint
f6-baecf2362d92548f
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, 1; 2: 8, 9, 2; 3: 40, 70, 36, 5; 4: 220, 525, 432, 140, 14; 5: 1274, 3861, 4400, 2310, 540, 42; 6: 7616, 28028, 40950, 30030, 11440, 2079, 132; 7: 46512, 201552, 359856, 340340, 182000, 54054, 8008, 429; 8: 288420, 1438965, 3038784, 3527160, 2441880, 1021020, 247520, 30888, 1430; 9: 1808950, 10214875, 24919488, 34321980, 29248296, 15872220, 5426400, 1108536, 119340, 4862;
%F FormulaThe row polynomials satisfy (8*n^4+4*n^3-8*n^2-4*n+(12*n^3-2*n^2-4*n)*x+(4*n^2+2*n)*x^2)*P(n,x) = (54*n^4-150*n^2+120*n-24+(48*n^4+73*n^3-232*n^2+153*n-30)*x+(2*n^4+68*n^3-67*n^2+17*n-2)*x^2+(3*n^3+18*n^2-20*n+5)*x^3+(n^2-2*n+1)*x^4)*P(n-1,x) - ((18*n^3+18*n^2-32*n+8)*x+(20*n^3+45*n^2-45*n+10)*x^2+(2*n^3+30*n^2-6*n-2)*x^3+(3*n^2+8*n-5)*x^4+(n-1)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t MathematicaP[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((54*n^4 - 150*n^2 + 120*n - 24 + (48*n^4 + 73*n^3 - 232*n^2 + 153*n - 30)*x + (2*n^4 + 68*n^3 - 67*n^2 + 17*n - 2)*x^2 + (3*n^3 + 18*n^2 - 20*n + 5)*x^3 + (n^2 - 2*n + 1)*x^4)*P[n - 1, x] - ((18*n^3 + 18*n^2 - 32*n + 8)*x + (20*n^3 + 45*n^2 - 45*n + 10)*x^2 + (2*n^3 + 30*n^2 - 6*n - 2)*x^3 + (3*n^2 + 8*n - 5)*x^4 + (n - 1)*x^5)*D[P[n - 1, x], x])/(8*n^4 + 4*n^3 - 8*n^2 - 4*n + (12*n^3 - 2*n^2 - 4*n)*x + (4*n^2 + 2*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((54*n^4 - 150*n^2 + 120*n - 24 + (48*n^4 + 73*n^3 - 232*n^2 + 153*n - 30)*x + (2*n^4 + 68*n^3 - 67*n^2 + 17*n - 2)*x^2 + (3*n^3 + 18*n^2 - 20*n + 5)*x^3 + (n^2 - 2*n + 1)*x^4)*v[i - 1] - ((18*n^3 + 18*n^2 - 32*n + 8)*x + (20*n^3 + 45*n^2 - 45*n + 10)*x^2 + (2*n^3 + 30*n^2 - 6*n - 2)*x^3 + (3*n^2 + 8*n - 5)*x^4 + (n - 1)*x^5)*deriv(v[i - 1], x))/(8*n^4 + 4*n^3 - 8*n^2 - 4*n + (12*n^3 - 2*n^2 - 4*n)*x + (4*n^2 + 2*n)*x^2)); 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 differential operators
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Proof chains ? JSON
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Laguerre’s theorem on differential operators [SP14] polytool ? step ›
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from
Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch th…?
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Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.?
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Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
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from
Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.?
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from
Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch th…?
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.