A391045 → reverse → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-2\,n - 6\,n x - 6\,n x^{2} - 2\,n x^{3} - 2\,n^{2} - 6\,n^{2} x - 6\,n^{2} x^{2} - 2\,n^{2} x^{3} + 4\,n^{3} + 12\,n^{3} x + 12\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n}(x) = \left(-24\,x - 132\,x^{2} - 288\,x^{3} - 312\,x^{4} - 168\,x^{5} - 36\,x^{6} - 2\,n + 44\,n x + 318\,n x^{2} + 790\,n x^{3} + 932\,n x^{4} + 534\,n x^{5} + 120\,n x^{6} - 2\,n^{2} - 13\,n^{2} x - 111\,n^{2} x^{2} - 356\,n^{2} x^{3} - 502\,n^{2} x^{4} - 327\,n^{2} x^{5} - 81\,n^{2} x^{6} + 4\,n^{3} + 11\,n^{3} x + 15\,n^{3} x^{2} + 34\,n^{3} x^{3} + 62\,n^{3} x^{4} + 51\,n^{3} x^{5} + 15\,n^{3} x^{6}\right)\,P_{n-1}(x) + \left(-12\,x - 102\,x^{2} - 324\,x^{3} - 516\,x^{4} - 444\,x^{5} - 198\,x^{6} - 36\,x^{7} + 18\,n x + 212\,n x^{2} + 770\,n x^{3} + 1320\,n x^{4} + 1190\,n x^{5} + 548\,n x^{6} + 102\,n x^{7} + 12\,n^{2} x + 22\,n^{2} x^{2} - 62\,n^{2} x^{3} - 228\,n^{2} x^{4} - 272\,n^{2} x^{5} - 146\,n^{2} x^{6} - 30\,n^{2} x^{7}\right)\,P_{n-1}'(x) + \left(-8\,x^{2} - 46\,x^{3} - 110\,x^{4} - 140\,x^{5} - 100\,x^{6} - 38\,x^{7} - 6\,x^{8} + 20\,n x^{2} + 115\,n x^{3} + 275\,n x^{4} + 350\,n x^{5} + 250\,n x^{6} + 95\,n x^{7} + 15\,n x^{8}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | 1 | 2 | 3 | |||||||||||||
| 3 | 1 | 6 | 6 | 13 | ||||||||||||
| 4 | 1 | 13 | 32 | 21 | 67 | |||||||||||
| 5 | 1 | 24 | 108 | 164 | 80 | 377 | ||||||||||
| 6 | 1 | 40 | 290 | 760 | 830 | 322 | 2243 | |||||||||
| 7 | 1 | 62 | 673 | 2692 | 4918 | 4184 | 1347 | 13877 | ||||||||
| 8 | 1 | 91 | 1407 | 8015 | 21700 | 30282 | 21077 | 5798 | 88371 | |||||||
| 9 | 1 | 128 | 2716 | 21056 | 78960 | 160272 | 180516 | 106248 | 25512 | 575409 | ||||||
| 10 | 1 | 174 | 4920 | 50244 | 249804 | 689904 | 1115676 | 1052100 | 536256 | 114236 | 3813315 | |||||
| 11 | 1 | 230 | 8460 | 111000 | 709380 | 2553768 | 5549670 | 7442640 | 6031800 | 2710520 | 518848 | 25636317 | ||||
| 12 | 1 | 297 | 13926 | 230109 | 1846746 | 8408862 | 23537052 | 42029262 | 48086478 | 34151942 | 13720806 | 2384538 | 174410019 | |||
| 13 | 1 | 376 | 22088 | 452100 | 4473535 | 25190528 | 88166848 | 200993144 | 304026250 | 303057040 | 191492268 | 69556368 | 11068567 | 1198499113 | ||
| 14 | 1 | 468 | 33930 | 848276 | 10195185 | 69760548 | 298650924 | 844066080 | 1618832358 | 2121402140 | 1872461448 | 1065375324 | 353096107 | 51817118 | 8306539907 | |
| 15 | 1 | 574 | 50687 | 1529164 | 22045023 | 180714534 | 930260331 | 3189463992 | 7533451926 | 12448976540 | 14379172808 | 11383596952 | 5889706277 | 1794776102 | 244370806 | 57998115717 |
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Technical details
- Node id
n-a91d6311c1cae769· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-cf0b07be82a54d68
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, 2; 3: 1, 6, 6; 4: 1, 13, 32, 21; 5: 1, 24, 108, 164, 80; 6: 1, 40, 290, 760, 830, 322; 7: 1, 62, 673, 2692, 4918, 4184, 1347; 8: 1, 91, 1407, 8015, 21700, 30282, 21077, 5798; 9: 1, 128, 2716, 21056, 78960, 160272, 180516, 106248, 25512; 10: 1, 174, 4920, 50244, 249804, 689904, 1115676, 1052100, 536256, 114236;
%F FormulaThe row polynomials satisfy (4*n^3-2*n^2-2*n+(12*n^3-6*n^2-6*n)*x+(12*n^3-6*n^2-6*n)*x^2+(4*n^3-2*n^2-2*n)*x^3)*P(n,x) = (4*n^3-2*n^2-2*n+(11*n^3-13*n^2+44*n-24)*x+(15*n^3-111*n^2+318*n-132)*x^2+(34*n^3-356*n^2+790*n-288)*x^3+(62*n^3-502*n^2+932*n-312)*x^4+(51*n^3-327*n^2+534*n-168)*x^5+(15*n^3-81*n^2+120*n-36)*x^6)*P(n-1,x) + ((12*n^2+18*n-12)*x+(22*n^2+212*n-102)*x^2-(62*n^2-770*n+324)*x^3-(228*n^2-1320*n+516)*x^4-(272*n^2-1190*n+444)*x^5-(146*n^2-548*n+198)*x^6-(30*n^2-102*n+36)*x^7)*d/dx P(n-1,x) + ((20*n-8)*x^2+(115*n-46)*x^3+(275*n-110)*x^4+(350*n-140)*x^5+(250*n-100)*x^6+(95*n-38)*x^7+(15*n-6)*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 2*n^2 - 2*n + (11*n^3 - 13*n^2 + 44*n - 24)*x + (15*n^3 - 111*n^2 + 318*n - 132)*x^2 + (34*n^3 - 356*n^2 + 790*n - 288)*x^3 + (62*n^3 - 502*n^2 + 932*n - 312)*x^4 + (51*n^3 - 327*n^2 + 534*n - 168)*x^5 + (15*n^3 - 81*n^2 + 120*n - 36)*x^6)*P[n - 1, x] + ((12*n^2 + 18*n - 12)*x + (22*n^2 + 212*n - 102)*x^2 - (62*n^2 - 770*n + 324)*x^3 - (228*n^2 - 1320*n + 516)*x^4 - (272*n^2 - 1190*n + 444)*x^5 - (146*n^2 - 548*n + 198)*x^6 - (30*n^2 - 102*n + 36)*x^7)*D[P[n - 1, x], x] + ((20*n - 8)*x^2 + (115*n - 46)*x^3 + (275*n - 110)*x^4 + (350*n - 140)*x^5 + (250*n - 100)*x^6 + (95*n - 38)*x^7 + (15*n - 6)*x^8)*D[P[n - 1, x], {x, 2}])/(4*n^3 - 2*n^2 - 2*n + (12*n^3 - 6*n^2 - 6*n)*x + (12*n^3 - 6*n^2 - 6*n)*x^2 + (4*n^3 - 2*n^2 - 2*n)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^3 - 2*n^2 - 2*n + (11*n^3 - 13*n^2 + 44*n - 24)*x + (15*n^3 - 111*n^2 + 318*n - 132)*x^2 + (34*n^3 - 356*n^2 + 790*n - 288)*x^3 + (62*n^3 - 502*n^2 + 932*n - 312)*x^4 + (51*n^3 - 327*n^2 + 534*n - 168)*x^5 + (15*n^3 - 81*n^2 + 120*n - 36)*x^6)*v[i - 1] + ((12*n^2 + 18*n - 12)*x + (22*n^2 + 212*n - 102)*x^2 - (62*n^2 - 770*n + 324)*x^3 - (228*n^2 - 1320*n + 516)*x^4 - (272*n^2 - 1190*n + 444)*x^5 - (146*n^2 - 548*n + 198)*x^6 - (30*n^2 - 102*n + 36)*x^7)*deriv(v[i - 1], x) + ((20*n - 8)*x^2 + (115*n - 46)*x^3 + (275*n - 110)*x^4 + (350*n - 140)*x^5 + (250*n - 100)*x^6 + (95*n - 38)*x^7 + (15*n - 6)*x^8)*deriv(deriv(v[i - 1], x), x))/(4*n^3 - 2*n^2 - 2*n + (12*n^3 - 6*n^2 - 6*n)*x + (12*n^3 - 6*n^2 - 6*n)*x^2 + (4*n^3 - 2*n^2 - 2*n)*x^3)); 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 [-1, 0]
- Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
reversal of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.?
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Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
reversal of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.?
- ↑ Pólya–Schur theorem on multiplier sequences (above)
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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: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.?
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from
reversal of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.?
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
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from
Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
reversal of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.?
Related ?
Same fingerprint ?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
- [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.
- [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023