A038329 → basis A013611 → basis A013610
nonnegative coefficientsonly rational roots
Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel
Recurrence
\(P_{n}(x) = \left(27 + 36\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 729 + 1944\,x + 1296\,x^{2}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | x15 | x16 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 729 | 1944 | 1296 | 3969 | ||||||||||||||
| 1 | 19683 | 78732 | 104976 | 46656 | 250047 | |||||||||||||
| 2 | 531441 | 2834352 | 5668704 | 5038848 | 1679616 | 15752961 | ||||||||||||
| 3 | 14348907 | 95659380 | 255091680 | 340122240 | 226748160 | 60466176 | 992436543 | |||||||||||
| 4 | 387420489 | 3099363912 | 10331213040 | 18366600960 | 18366600960 | 9795520512 | 2176782336 | 62523502209 | ||||||||||
| 5 | 10460353203 | 97629963228 | 390519852912 | 867821895360 | 1157095860480 | 925676688384 | 411411861504 | 78364164096 | 3938980639167 | |||||||||
| 6 | 282429536481 | 3012581722464 | 14058714704832 | 37489905879552 | 62483176465920 | 66648721563648 | 44432481042432 | 16926659444736 | 2821109907456 | 248155780267521 | ||||||||
| 7 | 7625597484987 | 91507169819844 | 488038239039168 | 1518341188121856 | 3036682376243712 | 4048909834991616 | 3599030964436992 | 2056589122535424 | 685529707511808 | 101559956668416 | 15633814156853823 | |||||||
| 8 | 205891132094649 | 2745215094595320 | 16471290567571920 | 58564588684700160 | 136650706930967040 | 218641131089547264 | 242934590099496960 | 185093021028188160 | 92546510514094080 | 27421188300472320 | 3656158440062976 | 984930291881790849 | ||||||
| 9 | 5559060566555523 | 81532888309481004 | 543552588729873360 | 2174210354919493440 | 5797894279785315840 | 10822735988932589568 | 14430314651910119424 | 13743156811342970880 | 9162104540895313920 | 4072046462620139520 | 1085879056698703872 | 131621703842267136 | 62050608388552823487 | |||||
| 10 | 150094635296999121 | 2401514164751985936 | 17611103874847896864 | 78271572777101763840 | 234814718331305291520 | 500938065773451288576 | 779236991203146448896 | 890556561375024513024 | 742130467812520427520 | 439781017962975068160 | 175912407185190027264 | 42645432044894552064 | 4738381338321616896 | 3909188328478827879681 | ||||
| 11 | 4052555153018976267 | 70244289318995588628 | 561954314551964709024 | 2747332204476271910784 | 9157774014920906369280 | 21978657635810175286272 | 39073169130329200508928 | 52097558840438934011904 | 52097558840438934011904 | 38590784326251062231040 | 20581751640667233189888 | 7484273323878993887232 | 1663171849750887530496 | 170581728179578208256 | 246278864694166156419903 | |||
| 12 | 109418989131512359209 | 2042487797121564038568 | 17701560908386888334256 | 94408324844730071116032 | 346163857764010260758784 | 923103620704027362023424 | 1846207241408054724046848 | 2813268177383702436642816 | 3282146206947652842749952 | 2917463295064580304666624 | 1944975530043053536444416 | 943018438808753229791232 | 314339479602917743263744 | 64479893251880562720768 | 6140942214464815497216 | 15515568475732467854453889 | ||
| 13 | 2954312706550833698643 | 59086254131016673972860 | 551471705222822290413360 | 3186280963509639900166080 | 12745123854038559600664320 | 37385696638513108161948672 | 83079325863362462582108160 | 142421701480049935855042560 | 189895601973399914473390080 | 196928772416859170564997120 | 157543017933487336451997696 | 95480616929386264516362240 | 42435829746393895340605440 | 13057178383505813950955520 | 2487081596858250276372480 | 221073919720733357899776 | 977480813971145474830595007 | |
| 14 | 79766443076872509863361 | 1701684118973280210418368 | 17016841189732802104183680 | 105882567402781879759365120 | 458824458745388145623915520 | 1468238267985242065996529664 | 3589026877297258383547072512 | 6836241671042396921042042880 | 10254362506563595381563064320 | 12153318526297594526296965120 | 11343097291211088224543834112 | 8249525302698973254213697536 | 4583069612610540696785387520 | 1880233687224837208937594880 | 537209624921382059696455680 | 95503933319356810612703232 | 7958661109946400884391936 | 61581291280182164914327485441 |
Compute more
?
Technical details
- Node id
n-d385989e7c5abdf8· revision 0 · JSON- Definition source
derived:basis_map_A013610- Fingerprint
f6-c91d5d506c107735
Observed on the stored rows ?
- Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
- 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): yes
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 729, 1944, 1296; 1: 19683, 78732, 104976, 46656; 2: 531441, 2834352, 5668704, 5038848, 1679616; 3: 14348907, 95659380, 255091680, 340122240, 226748160, 60466176; 4: 387420489, 3099363912, 10331213040, 18366600960, 18366600960, 9795520512, 2176782336; 5: 10460353203, 97629963228, 390519852912, 867821895360, 1157095860480, 925676688384, 411411861504, 78364164096; 6: 282429536481, 3012581722464, 14058714704832, 37489905879552, 62483176465920, 66648721563648, 44432481042432, 16926659444736, 2821109907456; 7: 7625597484987, 91507169819844, 488038239039168, 1518341188121856, 3036682376243712, 4048909834991616, 3599030964436992, 2056589122535424, 685529707511808, 101559956668416; 8: 205891132094649, 2745215094595320, 16471290567571920, 58564588684700160, 136650706930967040, 218641131089547264, 242934590099496960, 185093021028188160, 92546510514094080, 27421188300472320, 3656158440062976; 9: 5559060566555523, 81532888309481004, 543552588729873360, 2174210354919493440, 5797894279785315840, 10822735988932589568, 14430314651910119424, 13743156811342970880, 9162104540895313920, 4072046462620139520, 1085879056698703872, 131621703842267136;
%F FormulaThe row polynomials satisfy P(n,x) = (27+36*x)*P(n-1,x) for n >= 1, with P(0,x) = 729 + 1944*x + 1296*x^2.
%t MathematicaP[0, x_] := 729 + 1944*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 36*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] = 729 + 1944*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = (27 + 36*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]
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Coefficients of one sign?
Every nonzero row, n ≥ 0: 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 ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Proof chains ? JSON
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Geometric basis map (elementary) polytool ? step ›
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from
basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
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Geometric basis map (elementary) polytool ? step ›
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from
Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
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Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
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from
Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
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from
basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
Related ?
Same fingerprint ?
- A038330 → basis A013610
- A178821 → basis A007318 → basis A013611
- basis-map coefficients of A026729 → basis A013609 → basis A013609
- A038279 → basis A013610 → x ↦ x/(x+1)
- basis-map coefficients of A112227 → basis A013610 → basis A007318
- A136572 → basis A013610 → basis A007318
- A120910 → basis A013611
- A038284 → D
- A356546 → basis A013611 → basis A013609
- A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
- basis-map coefficients of A026729, shifted → basis A013609 → basis A013609
- A178820 → basis A013611 → basis A013609
- A038221 → basis A013611 → basis A013609
- A038330 → basis A013610 → D
- A196347 → x ↦ x+1 → basis A013611
- A038208 → basis A013611 → basis A013609
- A013610 → basis A013611 → x ↦ x+1
- basis-map coefficients of A112227 → basis A013609 → basis A013609
- basis-map coefficients of A026729 → basis A013610 → basis A007318
- A085880 → x ↦ x+1 → basis A013611
- A038335 → reverse
- A013609 → D → basis A013609
- A356546 → x ↦ x+1 → basis A013611
- A156992 → basis A013611 → basis A013609
- Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
- A130123 → basis A013609 → basis A013609
- A038334 → basis A013609 → basis A013610
- A027465 → x ↦ x/(x+1)
- A038329 → basis A013610 → basis A013609
- A120910 → basis A013611 → reverse
- A038243 → basis A013609 → x ↦ x/(x+1)
- A085881 → basis A013611 → basis A013609
- basis-map coefficients of A030528, shifted → basis A013610 → x ↦ x+1
- A013609 → basis A013609
- A130123 → basis A013610 → x ↦ x+1
- A120910 → basis A013611 → D
- A013609 → basis A013609 → D
- A120909 → basis A013610 → basis A013609
- A120910 → D → basis A013611
- basis-map coefficients of A030528, shifted → basis A013609 → basis A013609
- A038330 → basis A013610 → reverse
- A013610 → basis A007318
- A003506 → x ↦ x+1 → basis A013611
- A013610 → basis A007318 → D
- A038330 → x ↦ x/(x+1)
- A038335 → D
- A135065 → x ↦ x+1 → basis A013611
- basis-map coefficients of A026729, shifted → basis A013610 → x ↦ x+1
- A178822 → basis A013611 → basis A013609
- A094305 → x ↦ x+1 → basis A013611
- A038284 → reverse
- A027465 → D → basis A013610
- A028326 → basis A013611 → basis A013609
- A196347 → basis A013611 → basis A013609
- A178821 → basis A013611 → basis A013609
- A028326 → basis A007318 → basis A013611
- A178820 → basis A007318 → basis A013611
- A085880 → basis A013611 → basis A013609
- A178822 → x ↦ x+1 → basis A013611
References
- [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [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
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.