A099097 → basis A007318 → basis A013611
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(2 + 4\,x\right)\,P_{n-2}(x) + \left(6 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 12\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | ||||||||||||||
| 2 | 6 | 12 | 18 | |||||||||||||
| 3 | 38 | 148 | 144 | 330 | ||||||||||||
| 4 | 240 | 1392 | 2688 | 1728 | 6048 | |||||||||||
| 5 | 1516 | 11680 | 33712 | 43200 | 20736 | 110844 | ||||||||||
| 6 | 9576 | 92016 | 353376 | 677952 | 649728 | 248832 | 2031480 | |||||||||
| 7 | 60488 | 696432 | 3338592 | 8529472 | 12248064 | 9372672 | 2985984 | 37231704 | ||||||||
| 8 | 382080 | 5126784 | 29463552 | 94009344 | 179853312 | 206309376 | 131383296 | 35831808 | 682359552 | |||||||
| 9 | 2413456 | 36980480 | 247765632 | 948032000 | 2265846016 | 3463833600 | 3307474944 | 1803534336 | 429981696 | 12505862160 | ||||||
| 10 | 15244896 | 262626240 | 2009793792 | 8967252480 | 25707204096 | 49105185792 | 62498856960 | 51108102144 | 24365629440 | 5159780352 | 229199676192 | |||||
| 11 | 96296288 | 1842310976 | 15853730816 | 80808166912 | 270174074368 | 619108615168 | 984725655552 | 1073471864832 | 767565103104 | 325066162176 | 61917364224 | 4200629344416 | ||||
| 12 | 608267520 | 12795653376 | 122300209152 | 701067451392 | 2678025867264 | 7157779771392 | 13659075772416 | 18609750687744 | 17740216664064 | 11268960288768 | 4292937252864 | 743008370688 | 76986526256640 | |||
| 13 | 3842197696 | 88142937600 | 926425800960 | 5899038475264 | 25344545436672 | 77401902563328 | 172293697662976 | 281653259747328 | 335587325902848 | 284216754438144 | 162409246359552 | 56220966715392 | 8916100448256 | 1410961248686016 | ||
| 14 | 24269721216 | 602988374784 | 6912053088768 | 48402676202496 | 231016055863296 | 793573623631872 | 2018534287368192 | 3849299734904832 | 5503282508464128 | 5825847224696832 | 4438718247075840 | 2304894522359808 | 731120236756992 | 106993205379072 | 25859221633888128 | |
| 15 | 153302722688 | 4100821569280 | 50913602382336 | 388864474434560 | 2041213694384128 | 7789816399024128 | 22288284213370880 | 48570691167993856 | 81009079560437760 | 102905256262238208 | 98004161689288704 | 67856065017937920 | 32288171756617728 | 9451066475151360 | 1283918464548864 | 473931756902102400 |
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Technical details
- Node id
n-27830fbb0f09c824· revision 0 · JSON- Definition source
derived:basis_map_A013611- Fingerprint
f6-6fbed9d7a4fd25c7
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: 6, 12; 3: 38, 148, 144; 4: 240, 1392, 2688, 1728; 5: 1516, 11680, 33712, 43200, 20736; 6: 9576, 92016, 353376, 677952, 649728, 248832; 7: 60488, 696432, 3338592, 8529472, 12248064, 9372672, 2985984; 8: 382080, 5126784, 29463552, 94009344, 179853312, 206309376, 131383296, 35831808; 9: 2413456, 36980480, 247765632, 948032000, 2265846016, 3463833600, 3307474944, 1803534336, 429981696; 10: 15244896, 262626240, 2009793792, 8967252480, 25707204096, 49105185792, 62498856960, 51108102144, 24365629440, 5159780352;
%F FormulaThe row polynomials satisfy P(n,x) = (2+4*x)*P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 12*x.
%t MathematicaP[1, x_] := 1;
P[2, x_] := 6 + 12*x;
P[n_, x_] := P[n, x] = Expand[(2 + 4*x)*P[n - 2, x] + (6 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 12*x; for(n = 3, N, my(i = n); v[i] = (2 + 4*x)*v[i - 2] + (6 + 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, 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]
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Coefficients of one sign?
Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 1, 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 ≥ 2
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
basis_map_A007318 of Riordan array (1, 3+x).?
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Geometric basis map (elementary) polytool ? step ›
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from
Riordan array (1, 3+x).?
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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
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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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Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
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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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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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from
Riordan array (1, 3+x).?
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Geometric basis map (elementary) polytool ? step ›
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from
Riordan array (1, 3+x).?
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (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)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
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from
Riordan array (1, 3+x).?
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Geometric basis map (elementary) polytool ? step ›
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from
Riordan array (1, 3+x).?
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (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)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
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from
Riordan array (1, 3+x).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Riordan array (1, 3+x).?
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (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)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
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from
Riordan array (1, 3+x).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
-
from
Riordan array (1, 3+x).?
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (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)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
- ↑ Geometric basis map (elementary) (above)
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from
Riordan array (1, 3+x).?
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from
basis_map_A007318 of Riordan array (1, 3+x).?
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