A081581 → Veronese 2
nonnegative coefficients
Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means
Recurrence
\(P_{n}(x) = \left(2\right)\,P_{n-1}(x) + \left(-1 + x\right)\,P_{n-2}(x) + \left(12\,x\right)\,P_{n-3}(x) + \left(36\,x\right)\,P_{n-4}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)
\(P_{3}(x) = 1 + 15\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 1 | 1 | |||||||
| 2 | 1 | 1 | 2 | ||||||
| 3 | 1 | 15 | 16 | ||||||
| 4 | 1 | 78 | 1 | 80 | |||||
| 5 | 1 | 190 | 29 | 220 | |||||
| 6 | 1 | 351 | 351 | 1 | 704 | ||||
| 7 | 1 | 561 | 2339 | 43 | 2944 | ||||
| 8 | 1 | 820 | 9766 | 820 | 1 | 11408 | |||
| 9 | 1 | 1128 | 28806 | 9192 | 57 | 39184 | |||
| 10 | 1 | 1485 | 68034 | 68034 | 1485 | 1 | 139040 | ||
| 11 | 1 | 1891 | 138426 | 357078 | 23493 | 71 | 520960 | ||
| 12 | 1 | 2346 | 253359 | 1411404 | 253359 | 2346 | 1 | 1922816 | |
| 13 | 1 | 2850 | 428611 | 4457580 | 1987623 | 47986 | 85 | 6924736 | |
| 14 | 1 | 3403 | 682361 | 11867451 | 11867451 | 682361 | 3403 | 1 | 25106432 |
Computed in your browser ?
Technical details
- Node id
n-d1b3210357102aa3· revision 1 · JSON- Definition source
derived:veronese_section_2_0- Fingerprint
f6-d90ca2851160375c
Observed on the stored rows ?
- Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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
OEIS matches ?
No OEIS matches.
OEIS submission text ?
Fields to paste into an OEIS submission: the example %e, the formula
%F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows.
For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched;
the matched terms were compared with the published ones, which identifies the node with the entry on a
finite prefix, not for every n. The PARI/GP program is not machine-checked here.
The atlas's own rows: n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1; 2: 1, 1; 3: 1, 15; 4: 1, 78, 1; 5: 1, 190, 29; 6: 1, 351, 351, 1; 7: 1, 561, 2339, 43; 8: 1, 820, 9766, 820, 1; 9: 1, 1128, 28806, 9192, 57;
%F FormulaThe row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) + 12*x*P(n-3,x) + 36*x*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 1 + 15*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 1 + 15*x;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 2, x] + 12*x*P[n - 3, x] + 36*x*P[n - 4, x]];
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; v[3] = 1 + x; v[4] = 1 + 15*x; for(n = 4, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - x)*v[i - 2] + 12*x*v[i - 3] + 36*x*v[i - 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)Generated with polytool c48629374597.
Proved properties
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Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.by the cited steps:
e-8267555b73d20761 -
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).derived from the cited steps by a stated theorem:
e-8267555b73d20761 -
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.derived from the cited steps by a stated theorem:
e-8267555b73d20761 -
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.by the cited steps:
e-8267555b73d20761
Proof chains
1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON
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veronese_section r=2, j=0, shift 0
e-8267555b73d20761Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
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premise (sequence):
Pascal-(1,6,1) array.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
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theorem_check liu_wang_derivative_lag
e-dfce858c9d52dee3root: theorem checkLiu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
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preserver_image reversal, shift 0
e-5c5351a0cb2ba1dfAffine, reversal and Möbius substitutions (elementary) polytool ?
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premise (sequence):
Pascal-(1,6,1) array.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
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premise (sequence):
Pascal-(1,6,1) array.
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premise (sequence):
Pascal-(1,6,1) array.
Related sequences ?
No candidate equalities or related sequences.
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
- [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