A391533 → Veronese 2 (j=1) → (θ+1)²
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-192\,x - 96\,x^{2} - 60\,x^{3} + 12\,x^{4} + 64\,n + 48\,n x + 296\,n x^{2} - 138\,n x^{3} + 40\,n x^{4} - 32\,n^{2} x + 296\,n^{2} x^{2} - 144\,n^{2} x^{3} + 51\,n^{2} x^{4} - 160\,n^{3} x + 80\,n^{3} x^{2} - 102\,n^{3} x^{3} + 31\,n^{3} x^{4} + 32\,n^{4} x^{2} - 44\,n^{4} x^{3} + 9\,n^{4} x^{4} + 16\,n^{5} x^{2} - 8\,n^{5} x^{3} + n^{5} x^{4}\right)\,P_{n}(x) = \left(64 - 528\,x - 592\,x^{2} - 384\,x^{3} + 48\,x^{4} + 64\,n + 272\,n x + 1416\,n x^{2} - 60\,n x^{3} - 4\,n x^{4} + 36\,n x^{5} - 192\,n^{2} x + 200\,n^{2} x^{2} + 38\,n^{2} x^{3} - 88\,n^{2} x^{4} + 60\,n^{2} x^{5} - 160\,n^{3} x - 176\,n^{3} x^{2} + 10\,n^{3} x^{3} - 109\,n^{3} x^{4} + 37\,n^{3} x^{5} + 48\,n^{4} x^{2} + 20\,n^{4} x^{3} - 48\,n^{4} x^{4} + 10\,n^{4} x^{5} + 16\,n^{5} x^{2} + 8\,n^{5} x^{3} - 7\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(128\,x - 96\,x^{2} + 480\,x^{3} - 792\,x^{4} + 352\,x^{5} - 72\,x^{6} - 704\,n x^{2} + 1376\,n x^{3} - 1180\,n x^{4} + 628\,n x^{5} - 120\,n x^{6} - 320\,n^{2} x^{2} + 800\,n^{2} x^{3} - 860\,n^{2} x^{4} + 454\,n^{2} x^{5} - 74\,n^{2} x^{6} + 192\,n^{3} x^{3} - 320\,n^{3} x^{4} + 148\,n^{3} x^{5} - 20\,n^{3} x^{6} + 32\,n^{4} x^{3} - 48\,n^{4} x^{4} + 18\,n^{4} x^{5} - 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 4\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 4 | 4 | |||||||
| 1 | 8 | 4 | 12 | ||||||
| 2 | 12 | 24 | 36 | ||||||
| 3 | 16 | 76 | 9 | 101 | |||||
| 4 | 20 | 176 | 72 | 268 | |||||
| 5 | 24 | 340 | 306 | 16 | 686 | ||||
| 6 | 28 | 584 | 936 | 160 | 1708 | ||||
| 7 | 32 | 924 | 2331 | 848 | 25 | 4160 | |||
| 8 | 36 | 1376 | 5040 | 3200 | 300 | 9952 | |||
| 9 | 40 | 1956 | 9828 | 9696 | 1900 | 36 | 23456 | ||
| 10 | 44 | 2680 | 17712 | 25152 | 8500 | 504 | 54592 | ||
| 11 | 48 | 3564 | 29997 | 58080 | 30250 | 3708 | 49 | 125696 | |
| 12 | 52 | 4624 | 48312 | 122496 | 91300 | 19152 | 784 | 286720 | |
| 13 | 56 | 5876 | 74646 | 240240 | 243100 | 78156 | 6566 | 64 | 648704 |
| 14 | 60 | 7336 | 111384 | 443872 | 586300 | 268632 | 38416 | 1152 | 1457152 |
Compute more
?
Technical details
- Node id
n-beacdfa3ad6b4129· revision 0 · JSON- Definition source
derived:theta_plus_one_squared- Fingerprint
f6-75cc7c95cefa889b
Observed on the stored rows ?
- Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 4; 1: 8, 4; 2: 12, 24; 3: 16, 76, 9; 4: 20, 176, 72; 5: 24, 340, 306, 16; 6: 28, 584, 936, 160; 7: 32, 924, 2331, 848, 25; 8: 36, 1376, 5040, 3200, 300; 9: 40, 1956, 9828, 9696, 1900, 36;
%F FormulaThe row polynomials satisfy (64*n-(160*n^3+32*n^2-48*n+192)*x+(16*n^5+32*n^4+80*n^3+296*n^2+296*n-96)*x^2-(8*n^5+44*n^4+102*n^3+144*n^2+138*n+60)*x^3+(n^5+9*n^4+31*n^3+51*n^2+40*n+12)*x^4)*P(n,x) = (64*n+64-(160*n^3+192*n^2-272*n+528)*x+(16*n^5+48*n^4-176*n^3+200*n^2+1416*n-592)*x^2+(8*n^5+20*n^4+10*n^3+38*n^2-60*n-384)*x^3-(7*n^5+48*n^4+109*n^3+88*n^2+4*n-48)*x^4+(n^5+10*n^4+37*n^3+60*n^2+36*n)*x^5)*P(n-1,x) + (128*x-(320*n^2+704*n+96)*x^2+(32*n^4+192*n^3+800*n^2+1376*n+480)*x^3-(48*n^4+320*n^3+860*n^2+1180*n+792)*x^4+(18*n^4+148*n^3+454*n^2+628*n+352)*x^5-(2*n^4+20*n^3+74*n^2+120*n+72)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4.
%t MathematicaP[0, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((64*n + 64 - (160*n^3 + 192*n^2 - 272*n + 528)*x + (16*n^5 + 48*n^4 - 176*n^3 + 200*n^2 + 1416*n - 592)*x^2 + (8*n^5 + 20*n^4 + 10*n^3 + 38*n^2 - 60*n - 384)*x^3 - (7*n^5 + 48*n^4 + 109*n^3 + 88*n^2 + 4*n - 48)*x^4 + (n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n)*x^5)*P[n - 1, x] + (128*x - (320*n^2 + 704*n + 96)*x^2 + (32*n^4 + 192*n^3 + 800*n^2 + 1376*n + 480)*x^3 - (48*n^4 + 320*n^3 + 860*n^2 + 1180*n + 792)*x^4 + (18*n^4 + 148*n^3 + 454*n^2 + 628*n + 352)*x^5 - (2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x^6)*D[P[n - 1, x], x])/(64*n - (160*n^3 + 32*n^2 - 48*n + 192)*x + (16*n^5 + 32*n^4 + 80*n^3 + 296*n^2 + 296*n - 96)*x^2 - (8*n^5 + 44*n^4 + 102*n^3 + 144*n^2 + 138*n + 60)*x^3 + (n^5 + 9*n^4 + 31*n^3 + 51*n^2 + 40*n + 12)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 4; for(n = 1, N, my(i = n + 1); v[i] = ((64*n + 64 - (160*n^3 + 192*n^2 - 272*n + 528)*x + (16*n^5 + 48*n^4 - 176*n^3 + 200*n^2 + 1416*n - 592)*x^2 + (8*n^5 + 20*n^4 + 10*n^3 + 38*n^2 - 60*n - 384)*x^3 - (7*n^5 + 48*n^4 + 109*n^3 + 88*n^2 + 4*n - 48)*x^4 + (n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n)*x^5)*v[i - 1] + (128*x - (320*n^2 + 704*n + 96)*x^2 + (32*n^4 + 192*n^3 + 800*n^2 + 1376*n + 480)*x^3 - (48*n^4 + 320*n^3 + 860*n^2 + 1180*n + 792)*x^4 + (18*n^4 + 148*n^3 + 454*n^2 + 628*n + 352)*x^5 - (2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x^6)*deriv(v[i - 1], x))/(64*n - (160*n^3 + 32*n^2 - 48*n + 192)*x + (16*n^5 + 32*n^4 + 80*n^3 + 296*n^2 + 296*n - 96)*x^2 - (8*n^5 + 44*n^4 + 102*n^3 + 144*n^2 + 138*n + 60)*x^3 + (n^5 + 9*n^4 + 31*n^3 + 51*n^2 + 40*n + 12)*x^4)); 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]
-
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
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Proof chains ? JSON
-
Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
-
from
veronese_section_2_1 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n…?
-
Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
-
from
The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Iterated real-rootedness preservers (above)
- ↑ Parent-row provers of polytool (above)
-
from
The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
-
-
from
veronese_section_2_1 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n…?
References
- [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
- [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
- [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
- [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
- [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.