A056939 → θ+1 → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(24\,n + 72\,n x + 72\,n x^{2} + 24\,n x^{3} + 50\,n^{2} + 150\,n^{2} x + 150\,n^{2} x^{2} + 50\,n^{2} x^{3} + 35\,n^{3} + 105\,n^{3} x + 105\,n^{3} x^{2} + 35\,n^{3} x^{3} + 10\,n^{4} + 30\,n^{4} x + 30\,n^{4} x^{2} + 10\,n^{4} x^{3} + n^{5} + 3\,n^{5} x + 3\,n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n}(x) = \left(36\,x + 168\,x^{2} + 336\,x^{3} + 360\,x^{4} + 204\,x^{5} + 48\,x^{6} + 24\,n + 195\,n x + 542\,n x^{2} + 772\,n x^{3} + 654\,n x^{4} + 329\,n x^{5} + 76\,n x^{6} + 50\,n^{2} + 250\,n^{2} x + 329\,n^{2} x^{2} - 99\,n^{2} x^{3} - 521\,n^{2} x^{4} - 379\,n^{2} x^{5} - 86\,n^{2} x^{6} + 35\,n^{3} + 163\,n^{3} x + 187\,n^{3} x^{2} - 137\,n^{3} x^{3} - 428\,n^{3} x^{4} - 302\,n^{3} x^{5} - 70\,n^{3} x^{6} + 10\,n^{4} + 65\,n^{4} x + 169\,n^{4} x^{2} + 237\,n^{4} x^{3} + 197\,n^{4} x^{4} + 94\,n^{4} x^{5} + 20\,n^{4} x^{6} + n^{5} + 11\,n^{5} x + 45\,n^{5} x^{2} + 91\,n^{5} x^{3} + 98\,n^{5} x^{4} + 54\,n^{5} x^{5} + 12\,n^{5} x^{6}\right)\,P_{n-1}(x) + \left(36\,x + 228\,x^{2} + 648\,x^{3} + 1032\,x^{4} + 948\,x^{5} + 468\,x^{6} + 96\,x^{7} + 123\,n x + 715\,n x^{2} + 1878\,n x^{3} + 2822\,n x^{4} + 2503\,n x^{5} + 1215\,n x^{6} + 248\,n x^{7} + 108\,n^{2} x + 491\,n^{2} x^{2} + 960\,n^{2} x^{3} + 1090\,n^{2} x^{4} + 800\,n^{2} x^{5} + 363\,n^{2} x^{6} + 76\,n^{2} x^{7} + 33\,n^{3} x + 64\,n^{3} x^{2} - 138\,n^{3} x^{3} - 532\,n^{3} x^{4} - 623\,n^{3} x^{5} - 324\,n^{3} x^{6} - 64\,n^{3} x^{7} + 3\,n^{4} x - 12\,n^{4} x^{2} - 102\,n^{4} x^{3} - 228\,n^{4} x^{4} - 237\,n^{4} x^{5} - 120\,n^{4} x^{6} - 24\,n^{4} x^{7}\right)\,P_{n-1}'(x) + \left(12\,x^{2} + 84\,x^{3} + 240\,x^{4} + 360\,x^{5} + 300\,x^{6} + 132\,x^{7} + 24\,x^{8} + 37\,n x^{2} + 259\,n x^{3} + 740\,n x^{4} + 1110\,n x^{5} + 925\,n x^{6} + 407\,n x^{7} + 74\,n x^{8} + 28\,n^{2} x^{2} + 196\,n^{2} x^{3} + 560\,n^{2} x^{4} + 840\,n^{2} x^{5} + 700\,n^{2} x^{6} + 308\,n^{2} x^{7} + 56\,n^{2} x^{8} + 6\,n^{3} x^{2} + 42\,n^{3} x^{3} + 120\,n^{3} x^{4} + 180\,n^{3} x^{5} + 150\,n^{3} x^{6} + 66\,n^{3} x^{7} + 12\,n^{3} x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 1 | 3 | 4 | |||||||||||||
| 2 | 1 | 10 | 12 | 23 | ||||||||||||
| 3 | 1 | 23 | 73 | 55 | 152 | |||||||||||
| 4 | 1 | 44 | 276 | 504 | 276 | 1101 | ||||||||||
| 5 | 1 | 75 | 815 | 2705 | 3435 | 1477 | 8508 | |||||||||
| 6 | 1 | 118 | 2045 | 10940 | 24165 | 23442 | 8296 | 69007 | ||||||||
| 7 | 1 | 175 | 4557 | 36659 | 125111 | 205401 | 160951 | 48393 | 581248 | |||||||
| 8 | 1 | 248 | 9268 | 106904 | 527590 | 1302728 | 1695484 | 1113464 | 291010 | 5046697 | ||||||
| 9 | 1 | 339 | 17526 | 279594 | 1910496 | 6646920 | 12771150 | 13741170 | 7763160 | 1794320 | 44924676 | |||||
| 10 | 1 | 450 | 31230 | 669360 | 6135570 | 28840140 | 76666800 | 120132000 | 110043000 | 54532400 | 11297760 | 408348711 | ||||
| 11 | 1 | 583 | 52965 | 1488795 | 17859600 | 110082720 | 388452240 | 830900400 | 1096976760 | 874212680 | 385757240 | 72413640 | 3778197624 | |||
| 12 | 1 | 740 | 86152 | 3111328 | 47865312 | 378183168 | 1720707912 | 4821765168 | 8600738784 | 9798098816 | 6906972424 | 2746521296 | 471309944 | 35495361045 | ||
| 13 | 1 | 923 | 135213 | 6163443 | 119530125 | 1188720819 | 6823529427 | 24341062317 | 56303950131 | 85957785485 | 86047724191 | 54363563553 | 19671173223 | 3108745785 | 337932084636 | |
| 14 | 1 | 1134 | 205751 | 11655644 | 280741461 | 3461307850 | 24639780165 | 109532880600 | 319333959945 | 627123767270 | 835754782863 | 745726225972 | 426757016413 | 141657380802 | 20746732688 | 3255026438559 |
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Technical details
- Node id
n-2500f91b4fa8e65f· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-7430f58d15347800
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, 3; 2: 1, 10, 12; 3: 1, 23, 73, 55; 4: 1, 44, 276, 504, 276; 5: 1, 75, 815, 2705, 3435, 1477; 6: 1, 118, 2045, 10940, 24165, 23442, 8296; 7: 1, 175, 4557, 36659, 125111, 205401, 160951, 48393; 8: 1, 248, 9268, 106904, 527590, 1302728, 1695484, 1113464, 291010; 9: 1, 339, 17526, 279594, 1910496, 6646920, 12771150, 13741170, 7763160, 1794320;
%F FormulaThe row polynomials satisfy (n^5+10*n^4+35*n^3+50*n^2+24*n+(3*n^5+30*n^4+105*n^3+150*n^2+72*n)*x+(3*n^5+30*n^4+105*n^3+150*n^2+72*n)*x^2+(n^5+10*n^4+35*n^3+50*n^2+24*n)*x^3)*P(n,x) = (n^5+10*n^4+35*n^3+50*n^2+24*n+(11*n^5+65*n^4+163*n^3+250*n^2+195*n+36)*x+(45*n^5+169*n^4+187*n^3+329*n^2+542*n+168)*x^2+(91*n^5+237*n^4-137*n^3-99*n^2+772*n+336)*x^3+(98*n^5+197*n^4-428*n^3-521*n^2+654*n+360)*x^4+(54*n^5+94*n^4-302*n^3-379*n^2+329*n+204)*x^5+(12*n^5+20*n^4-70*n^3-86*n^2+76*n+48)*x^6)*P(n-1,x) + ((3*n^4+33*n^3+108*n^2+123*n+36)*x-(12*n^4-64*n^3-491*n^2-715*n-228)*x^2-(102*n^4+138*n^3-960*n^2-1878*n-648)*x^3-(228*n^4+532*n^3-1090*n^2-2822*n-1032)*x^4-(237*n^4+623*n^3-800*n^2-2503*n-948)*x^5-(120*n^4+324*n^3-363*n^2-1215*n-468)*x^6-(24*n^4+64*n^3-76*n^2-248*n-96)*x^7)*d/dx P(n-1,x) + ((6*n^3+28*n^2+37*n+12)*x^2+(42*n^3+196*n^2+259*n+84)*x^3+(120*n^3+560*n^2+740*n+240)*x^4+(180*n^3+840*n^2+1110*n+360)*x^5+(150*n^3+700*n^2+925*n+300)*x^6+(66*n^3+308*n^2+407*n+132)*x^7+(12*n^3+56*n^2+74*n+24)*x^8)*d^2/dx^2 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[((n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (11*n^5 + 65*n^4 + 163*n^3 + 250*n^2 + 195*n + 36)*x + (45*n^5 + 169*n^4 + 187*n^3 + 329*n^2 + 542*n + 168)*x^2 + (91*n^5 + 237*n^4 - 137*n^3 - 99*n^2 + 772*n + 336)*x^3 + (98*n^5 + 197*n^4 - 428*n^3 - 521*n^2 + 654*n + 360)*x^4 + (54*n^5 + 94*n^4 - 302*n^3 - 379*n^2 + 329*n + 204)*x^5 + (12*n^5 + 20*n^4 - 70*n^3 - 86*n^2 + 76*n + 48)*x^6)*P[n - 1, x] + ((3*n^4 + 33*n^3 + 108*n^2 + 123*n + 36)*x - (12*n^4 - 64*n^3 - 491*n^2 - 715*n - 228)*x^2 - (102*n^4 + 138*n^3 - 960*n^2 - 1878*n - 648)*x^3 - (228*n^4 + 532*n^3 - 1090*n^2 - 2822*n - 1032)*x^4 - (237*n^4 + 623*n^3 - 800*n^2 - 2503*n - 948)*x^5 - (120*n^4 + 324*n^3 - 363*n^2 - 1215*n - 468)*x^6 - (24*n^4 + 64*n^3 - 76*n^2 - 248*n - 96)*x^7)*D[P[n - 1, x], x] + ((6*n^3 + 28*n^2 + 37*n + 12)*x^2 + (42*n^3 + 196*n^2 + 259*n + 84)*x^3 + (120*n^3 + 560*n^2 + 740*n + 240)*x^4 + (180*n^3 + 840*n^2 + 1110*n + 360)*x^5 + (150*n^3 + 700*n^2 + 925*n + 300)*x^6 + (66*n^3 + 308*n^2 + 407*n + 132)*x^7 + (12*n^3 + 56*n^2 + 74*n + 24)*x^8)*D[P[n - 1, x], {x, 2}])/(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (3*n^5 + 30*n^4 + 105*n^3 + 150*n^2 + 72*n)*x + (3*n^5 + 30*n^4 + 105*n^3 + 150*n^2 + 72*n)*x^2 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*x^3)]];
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] = ((n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (11*n^5 + 65*n^4 + 163*n^3 + 250*n^2 + 195*n + 36)*x + (45*n^5 + 169*n^4 + 187*n^3 + 329*n^2 + 542*n + 168)*x^2 + (91*n^5 + 237*n^4 - 137*n^3 - 99*n^2 + 772*n + 336)*x^3 + (98*n^5 + 197*n^4 - 428*n^3 - 521*n^2 + 654*n + 360)*x^4 + (54*n^5 + 94*n^4 - 302*n^3 - 379*n^2 + 329*n + 204)*x^5 + (12*n^5 + 20*n^4 - 70*n^3 - 86*n^2 + 76*n + 48)*x^6)*v[i - 1] + ((3*n^4 + 33*n^3 + 108*n^2 + 123*n + 36)*x - (12*n^4 - 64*n^3 - 491*n^2 - 715*n - 228)*x^2 - (102*n^4 + 138*n^3 - 960*n^2 - 1878*n - 648)*x^3 - (228*n^4 + 532*n^3 - 1090*n^2 - 2822*n - 1032)*x^4 - (237*n^4 + 623*n^3 - 800*n^2 - 2503*n - 948)*x^5 - (120*n^4 + 324*n^3 - 363*n^2 - 1215*n - 468)*x^6 - (24*n^4 + 64*n^3 - 76*n^2 - 248*n - 96)*x^7)*deriv(v[i - 1], x) + ((6*n^3 + 28*n^2 + 37*n + 12)*x^2 + (42*n^3 + 196*n^2 + 259*n + 84)*x^3 + (120*n^3 + 560*n^2 + 740*n + 240)*x^4 + (180*n^3 + 840*n^2 + 1110*n + 360)*x^5 + (150*n^3 + 700*n^2 + 925*n + 300)*x^6 + (66*n^3 + 308*n^2 + 407*n + 132)*x^7 + (12*n^3 + 56*n^2 + 74*n + 24)*x^8)*deriv(deriv(v[i - 1], x), x))/(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (3*n^5 + 30*n^4 + 105*n^3 + 150*n^2 + 72*n)*x + (3*n^5 + 30*n^4 + 105*n^3 + 150*n^2 + 72*n)*x^2 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*x^3)); 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 [-1, 0]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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
theta_plus_one of Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane parti…?
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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
Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <=…?
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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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from
Triangle of Dyck paths counted by number of long interior inclines.?
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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 of Dyck paths counted by number of long interior inclines.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <=…?
- ↑ Pólya–Schur theorem on multiplier sequences (above)
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <=…?
- ↑ 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
Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <=…?
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
-
from
Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <=…?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Row-hypergeometric rows (multiplier sequences) (above)
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from
Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <=…?
-
Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
-
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from
theta_plus_one of Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane parti…?
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