A128100 → Veronese 2 (j=1) → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(x\right)\,P_{n-4}(x) + \left(-2\right)\,P_{n-3}(x) + \left(1\right)\,P_{n-2}(x) + \left(2\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)
\(P_{3}(x) = 2\)
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 0 | ||||
| 1 | 0 | ||||
| 2 | 1 | 1 | |||
| 3 | 2 | 2 | |||
| 4 | 5 | 5 | |||
| 5 | 10 | 10 | |||
| 6 | 21 | 1 | 22 | ||
| 7 | 42 | 4 | 46 | ||
| 8 | 85 | 14 | 99 | ||
| 9 | 170 | 40 | 210 | ||
| 10 | 341 | 107 | 1 | 449 | |
| 11 | 682 | 268 | 6 | 956 | |
| 12 | 1365 | 648 | 27 | 2040 | |
| 13 | 2730 | 1520 | 98 | 4348 | |
| 14 | 5461 | 3493 | 318 | 1 | 9273 |
Compute more
?
Technical details
- Node id
n-a814bc28eb725e07· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f5-bf00ef2587637288
Observed on the stored rows ?
- Degrees: -1, -1, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3 (irregular)
- 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: 0; 1: 0; 2: 1; 3: 2; 4: 5; 5: 10; 6: 21, 1; 7: 42, 4; 8: 85, 14; 9: 170, 40;
%F FormulaThe row polynomials satisfy P(n,x) = x*P(n-4,x) - 2*P(n-3,x) + P(n-2,x) + 2*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1, P(3,x) = 2.
%t MathematicaP[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 2;
P[n_, x_] := P[n, x] = Expand[x*P[n - 4, x] - 2*P[n - 3, x] + P[n - 2, x] + 2*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; v[4] = 2; for(n = 4, N, my(i = n + 1); v[i] = x*v[i - 4] - 2*v[i - 3] + v[i - 2] + 2*v[i - 1]); 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]
- 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
veronese_section_2_1 of Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of …?
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Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
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from
Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pie…?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
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Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
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from
delannoy?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
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from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
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from
delannoy?
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
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from
delannoy?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
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from
delannoy?
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from
Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pie…?
-
Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
-
from
Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pie…?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
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from
Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pie…?
-
-
from
veronese_section_2_1 of Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of …?
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