A030528 → basis A013611 → basis A013610
nonnegative coefficients
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)
Recurrence
\(P_{n}(x) = \left(5 + 12\,x\right)\,P_{n-2}(x) + \left(5 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 12\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 6 | 12 | 18 | |||||||||||||
| 2 | 35 | 144 | 144 | 323 | ||||||||||||
| 3 | 205 | 1272 | 2592 | 1728 | 5797 | |||||||||||
| 4 | 1200 | 9960 | 30672 | 41472 | 20736 | 104040 | ||||||||||
| 5 | 7025 | 73020 | 301104 | 615168 | 622080 | 248832 | 1867229 | |||||||||
| 6 | 41125 | 513600 | 2654640 | 7264512 | 11093760 | 8957952 | 2985984 | 33511573 | ||||||||
| 7 | 240750 | 3510900 | 21818160 | 74867328 | 153135360 | 186624000 | 125411328 | 35831808 | 601439634 | |||||||
| 8 | 1409375 | 23505000 | 170658000 | 704332800 | 1806727680 | 2948659200 | 2988969984 | 1719926784 | 429981696 | 10794170519 | ||||||
| 9 | 8250625 | 154881000 | 1286571600 | 6205714560 | 19149716736 | 39194772480 | 53195304960 | 46151368704 | 23219011584 | 5159780352 | 193725372601 | |||||
| 10 | 48300000 | 1007850000 | 9426780000 | 52036992000 | 187702790400 | 462194491392 | 786642554880 | 913567776768 | 692700512256 | 309586821120 | 61917364224 | 3476832233040 | ||||
| 11 | 282753125 | 6492262500 | 67519530000 | 419773752000 | 1733175014400 | 4989176404992 | 10215860465664 | 14876650045440 | 15096227364864 | 10164767293440 | 4086546038784 | 743008370688 | 62399479295897 | |||
| 12 | 1655265625 | 41473200000 | 474732900000 | 3282409440000 | 15266117952000 | 50307388139520 | 120428965859328 | 210981125357568 | 268427253252096 | 241838905098240 | 146434566389760 | 53496602689536 | 8916100448256 | 1119897295991929 | ||
| 13 | 9690093750 | 263083537500 | 3286847700000 | 25017945120000 | 129422663136000 | 480474338319360 | 1316782906159104 | 2697026792914944 | 4127910707920896 | 4662300129361920 | 3776649630842880 | 2077451404443648 | 695455834963968 | 106993205379072 | 20099045179893042 | |
| 14 | 56726796875 | 1658928000000 | 22462583850000 | 186640740000000 | 1063048160160000 | 4390174005350400 | 13555440077598720 | 31786582055583744 | 56877784825135104 | 77276750706376704 | 78465089399685120 | 57731750402457600 | 29093235762659328 | 8987429251842048 | 1283918464548864 | 360722022090044507 |
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Technical details
- Node id
n-2e8faceb378aff28· revision 0 · JSON- Definition source
derived:basis_map_A013610- Fingerprint
f6-42566bedfc909616
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: 6, 12; 2: 35, 144, 144; 3: 205, 1272, 2592, 1728; 4: 1200, 9960, 30672, 41472, 20736; 5: 7025, 73020, 301104, 615168, 622080, 248832; 6: 41125, 513600, 2654640, 7264512, 11093760, 8957952, 2985984; 7: 240750, 3510900, 21818160, 74867328, 153135360, 186624000, 125411328, 35831808; 8: 1409375, 23505000, 170658000, 704332800, 1806727680, 2948659200, 2988969984, 1719926784, 429981696; 9: 8250625, 154881000, 1286571600, 6205714560, 19149716736, 39194772480, 53195304960, 46151368704, 23219011584, 5159780352;
%F FormulaThe row polynomials satisfy P(n,x) = (5+12*x)*P(n-2,x) + (5+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 12*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 6 + 12*x;
P[n_, x_] := P[n, x] = Expand[(5 + 12*x)*P[n - 2, x] + (5 + 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] = 6 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (5 + 12*x)*v[i - 2] + (5 + 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_A013611 of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Geometric basis map (elementary) polytool ? step ›
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from
basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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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) = binomial(k,n-k).?
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from
basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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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
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.?
-
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-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
-
from
a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
- ↑ Ma–Wang interval theorem (above)
-
-
from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
-
Geometric basis map (elementary) polytool ? step ›
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
- ↑ 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)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
-
Geometric basis map (elementary) polytool ? step ›
-
from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
- ↑ 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)
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
- ↑ Affine-current basis map theorem (above)
- ↑ Ma–Wang interval theorem (above)
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from
Triangle read by rows: T(n,k) = binomial(k,n-k).?
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from
basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(k,n-k).?
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