A026729 → basis A013609 → reverse
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-4 - 14\,x - 10\,x^{2} + 4\,n + 8\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 12\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | 2 | 1 | 3 | |||||||||||||
| 3 | 4 | 6 | 2 | 12 | ||||||||||||
| 4 | 8 | 20 | 14 | 3 | 45 | |||||||||||
| 5 | 16 | 56 | 64 | 30 | 5 | 171 | ||||||||||
| 6 | 32 | 144 | 232 | 172 | 60 | 8 | 648 | |||||||||
| 7 | 64 | 352 | 736 | 760 | 416 | 116 | 13 | 2457 | ||||||||
| 8 | 128 | 832 | 2144 | 2864 | 2168 | 940 | 218 | 21 | 9315 | |||||||
| 9 | 256 | 1920 | 5888 | 9696 | 9456 | 5640 | 2024 | 402 | 34 | 35316 | ||||||
| 10 | 512 | 4352 | 15488 | 30400 | 36480 | 27936 | 13736 | 4204 | 730 | 55 | 133893 | |||||
| 11 | 1024 | 9728 | 39424 | 89984 | 128640 | 120960 | 76144 | 31832 | 8492 | 1310 | 89 | 507627 | ||||
| 12 | 2048 | 21504 | 97792 | 254720 | 423552 | 473920 | 365600 | 195216 | 70960 | 16776 | 2328 | 144 | 1924560 | |||
| 13 | 4096 | 47104 | 237568 | 695808 | 1321216 | 1718656 | 1575680 | 1029280 | 476944 | 153328 | 32544 | 4104 | 233 | 7296561 | ||
| 14 | 8192 | 102400 | 567296 | 1846272 | 3945472 | 5860352 | 6241408 | 4839360 | 2739200 | 1120736 | 322928 | 62184 | 7186 | 377 | 27663363 | |
| 15 | 16384 | 221184 | 1335296 | 4782080 | 11366400 | 19004416 | 23101696 | 20790144 | 13952000 | 6963840 | 2550192 | 665712 | 117308 | 12510 | 610 | 104879772 |
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Technical details
- Node id
n-90d459f6a295699a· revision 0 · JSON- Definition source
derived:reversal- Fingerprint
f6-ab1d67dd1b7f53fc
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 -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: 2, 1; 3: 4, 6, 2; 4: 8, 20, 14, 3; 5: 16, 56, 64, 30, 5; 6: 32, 144, 232, 172, 60, 8; 7: 64, 352, 736, 760, 416, 116, 13; 8: 128, 832, 2144, 2864, 2168, 940, 218, 21; 9: 256, 1920, 5888, 9696, 9456, 5640, 2024, 402, 34; 10: 512, 4352, 15488, 30400, 36480, 27936, 13736, 4204, 730, 55;
%F FormulaThe row polynomials satisfy (2*n-2)*P(n,x) = (4*n-4+(8*n-14)*x+(5*n-10)*x^2)*P(n-1,x) - (4*x+12*x^2+5*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t MathematicaP[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (8*n - 14)*x + (5*n - 10)*x^2)*P[n - 1, x] - (4*x + 12*x^2 + 5*x^3)*D[P[n - 1, x], x])/(2*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n - 4 + (8*n - 14)*x + (5*n - 10)*x^2)*v[i - 1] - (4*x + 12*x^2 + 5*x^3)*deriv(v[i - 1], x))/(2*n - 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 (−∞, 0]
- 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
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
basis_map_A013609 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward ant…?
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Geometric basis map (elementary) polytool ? step ›
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from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
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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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Geometric basis map (elementary) polytool ? step ›
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from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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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.?
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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
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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.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
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from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
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from
basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
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Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›
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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)
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from
Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
-
Geometric basis map (elementary) polytool ? step ›
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Geometric basis map (elementary) polytool ? step ›
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from
basis_map_A013609 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward ant…?
Related ?
Same fingerprint ?
References
- [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