A026729 → Veronese 2 → x ↦ x+1
nonnegative coefficients
Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(1 + x\right)\,P_{n-4}(x) + \left(2 + 2\,x\right)\,P_{n-3}(x) + \left(1 + x\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 1 + x\)
\(P_{4}(x) = 2 + 2\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | |||||||
| 2 | 0 | ||||||||
| 3 | 1 | 1 | 2 | ||||||
| 4 | 2 | 2 | 4 | ||||||
| 5 | 2 | 3 | 1 | 6 | |||||
| 6 | 4 | 8 | 4 | 16 | |||||
| 7 | 7 | 15 | 9 | 1 | 32 | ||||
| 8 | 10 | 26 | 22 | 6 | 64 | ||||
| 9 | 17 | 51 | 52 | 19 | 1 | 140 | |||
| 10 | 28 | 92 | 108 | 52 | 8 | 288 | |||
| 11 | 44 | 162 | 223 | 137 | 33 | 1 | 600 | ||
| 12 | 72 | 292 | 454 | 330 | 106 | 10 | 1264 | ||
| 13 | 117 | 514 | 888 | 751 | 310 | 51 | 1 | 2632 | |
| 14 | 188 | 896 | 1716 | 1664 | 836 | 192 | 12 | 5504 | |
| 15 | 305 | 1565 | 3279 | 3567 | 2103 | 627 | 73 | 1 | 11520 |
Compute more
?
Technical details
- Node id
n-e47f6c059bb35636· revision 0 · JSON- Definition source
derived:shift_x_plus_one- Fingerprint
f6-724a7ffe5bb049d0
Observed on the stored rows ?
- Degrees: 0, -1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 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 = 1, …, ascending powers of x
%e ExampleTriangle begins: 1: 1; 2: 0; 3: 1, 1; 4: 2, 2; 5: 2, 3, 1; 6: 4, 8, 4; 7: 7, 15, 9, 1; 8: 10, 26, 22, 6; 9: 17, 51, 52, 19, 1; 10: 28, 92, 108, 52, 8;
%F FormulaThe row polynomials satisfy P(n,x) = (1+x)*P(n-4,x) + (2+2*x)*P(n-3,x) + (1+x)*P(n-2,x) for n >= 5, with P(1,x) = 1, P(2,x) = 0, P(3,x) = 1 + x, P(4,x) = 2 + 2*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 0;
P[3, x_] := 1 + x;
P[4, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 4, x] + (2 + 2*x)*P[n - 3, x] + (1 + x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 0; v[3] = 1 + x; v[4] = 2 + 2*x; for(n = 5, N, my(i = n); v[i] = (1 + x)*v[i - 4] + (2 + 2*x)*v[i - 3] + (1 + x)*v[i - 2]); 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]
- 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
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
veronese_section_2_0 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward …?
-
Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
-
from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
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…?
-
Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
-
from
delannoy?
-
Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
delannoy?
- ↑ Liu–Wang theorem with a lag row (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
-
Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
-
Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
no recorded property covers this requirement
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
-
from
delannoy?
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
-
from
delannoy?
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Affine-current basis map theorem (above)
-
-
from
delannoy?
-
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…?
-
Geometric basis map (elementary) polytool ? step ›
-
from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.?
-
Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
-
-
from
Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
-
Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
-
-
from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
-
-
from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
-
Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
-
from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
no recorded property covers this requirement
-
from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
- ↑ Ma–Wang interval theorem (above)
-
-
from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
-
-
from
veronese_section_2_0 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward …?
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
- [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741