A113953 → basis A013610 → basis A013611
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(8 + 24\,x\right)\,P_{n-2}(x) + \left(4 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 12\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 4 | 12 | 16 | |||||||||||||
| 2 | 24 | 120 | 144 | 288 | ||||||||||||
| 3 | 128 | 960 | 2304 | 1728 | 5120 | |||||||||||
| 4 | 704 | 6912 | 24768 | 38016 | 20736 | 91136 | ||||||||||
| 5 | 3840 | 46848 | 223488 | 518400 | 580608 | 248832 | 1622016 | |||||||||
| 6 | 20992 | 305664 | 1820160 | 5654016 | 9621504 | 8460288 | 2985984 | 28868608 | ||||||||
| 7 | 114688 | 1941504 | 13860864 | 53968896 | 123420672 | 165224448 | 119439360 | 35831808 | 513802240 | |||||||
| 8 | 626688 | 12091392 | 100638720 | 471121920 | 1353977856 | 2440544256 | 2687385600 | 1648263168 | 429981696 | 9144631296 | ||||||
| 9 | 3424256 | 74170368 | 705134592 | 3856564224 | 13351993344 | 30293803008 | 44956975104 | 41994878976 | 22359048192 | 5159780352 | 162755772416 | |||||
| 10 | 18710528 | 449544192 | 4805885952 | 30072176640 | 121825492992 | 333418954752 | 623425683456 | 785146576896 | 636372910080 | 299267260416 | 61917364224 | 2896720560128 | ||||
| 11 | 102236160 | 2698248192 | 32039239680 | 225735081984 | 1047541579776 | 3358379999232 | 7581437263872 | 12036620943360 | 13154000044032 | 9411439362048 | 3962711310336 | 743008370688 | 51555713679360 | |||
| 12 | 558628864 | 16065232896 | 209772085248 | 1643329880064 | 8595323486208 | 31595182424064 | 83615769427968 | 160367119958016 | 220989952622592 | 213160845901824 | 136465870749696 | 52010585948160 | 8916100448256 | 917586476793856 | ||
| 13 | 3052404736 | 95004131328 | 1352943009792 | 11665406951424 | 67899227111424 | 281532649439232 | 854857884893184 | 1923105174847488 | 3202476152979456 | 3895510331031552 | 3361369868992512 | 1946681931202560 | 677623634067456 | 106993205379072 | 16331166466441216 | |
| 14 | 16678649856 | 558574338048 | 8615563886592 | 81078113009664 | 519784296873984 | 2399970546155520 | 8225033866444800 | 21240430744043520 | 41503897210060800 | 61020800790036480 | 66399190715990016 | 51814431738298368 | 27390260577042432 | 8773442841083904 | 1283918464548864 | 290661430720462848 |
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Technical details
- Node id
n-72b8b9b969b0952b· revision 0 · JSON- Definition source
derived:basis_map_A013611- Fingerprint
f6-3f5c634f6f648d33
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 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
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 4, 12; 2: 24, 120, 144; 3: 128, 960, 2304, 1728; 4: 704, 6912, 24768, 38016, 20736; 5: 3840, 46848, 223488, 518400, 580608, 248832; 6: 20992, 305664, 1820160, 5654016, 9621504, 8460288, 2985984; 7: 114688, 1941504, 13860864, 53968896, 123420672, 165224448, 119439360, 35831808; 8: 626688, 12091392, 100638720, 471121920, 1353977856, 2440544256, 2687385600, 1648263168, 429981696; 9: 3424256, 74170368, 705134592, 3856564224, 13351993344, 30293803008, 44956975104, 41994878976, 22359048192, 5159780352;
%F FormulaThe row polynomials satisfy P(n,x) = (8+24*x)*P(n-2,x) + (4+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 12*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 4 + 12*x;
P[n_, x_] := P[n, x] = Expand[(8 + 24*x)*P[n - 2, x] + (4 + 12*x)*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] = 1; v[2] = 4 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 24*x)*v[i - 2] + (4 + 12*x)*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 (−∞, 0]
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
basis_map_A013610 of A Jacobsthal triangle.?
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Geometric basis map (elementary) polytool ? step ›
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from
A Jacobsthal triangle.?
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Geometric basis map (elementary) polytool ? step ›
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from
basis-map coefficients of A Jacobsthal triangle.?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
A Jacobsthal triangle.?
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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
A Jacobsthal triangle.?
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from
basis-map coefficients of A Jacobsthal triangle.?
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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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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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Geometric basis map (elementary) polytool ? step ›
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pi…?
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Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
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from
Pascal-(1,2,1) array read by antidiagonals.?
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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 ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
Pascal-(1,2,1) array read by antidiagonals.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
A Jacobsthal triangle.?
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
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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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Geometric basis map (elementary) polytool ? step ›
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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.?
- ↑ Ma–Wang interval theorem (above)
-
from
A Jacobsthal triangle.?
-
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
A Jacobsthal triangle.?
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
-
from
A Jacobsthal triangle.?
-
from
Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pi…?
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
-
-
from
A Jacobsthal triangle.?
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Geometric basis map (elementary) polytool ? step ›
-
from
A Jacobsthal triangle.?
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
-
from
A Jacobsthal triangle.?
-
Geometric basis map (elementary) polytool ? step ›
-
from
A Jacobsthal triangle.?
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Liu–Wang theorem for three-term recurrences (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
-
from
A Jacobsthal triangle.?
-
-
from
basis_map_A013610 of A Jacobsthal triangle.?
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