Real-rooted atlas

A038336 → x ↦ x+1 → basis A013609

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(32 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1024 + 1280\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0102412804002704
13276861440384008000140608
210485762621440245760010240001600007311616
33355443210485760013107200081920000256000003200000380204032
4107374182440265318406291456000524288000024576000006144000006400000019770609664
534359738368150323855360281857228800293601280000183500800000688128000001433600000012800000001028071702528
610995116277765497558138880120259084288001503238553600011744051200000587202560000018350080000003276800000002560000000053459728531456
7351843720888321979120929996804947802324992007215545057280006764573491200004227858432000001761607680000004718592000000073728000000005120000000002779905883635712
81125899906842624703687441776640019791209299968000329853488332800003607772528640000027058293964800000140928614400000005033164800000000117964800000000016384000000000010240000000000144555105949057024
9360287970189639682476979795053772807740561859543040001451355348664320000181419418583040000015874199126016000009921374453760000004429185024000000001384120320000000002883584000000000036044800000000002048000000000007516865509350965248
1011529215046068469768646911284551352320297237575406452736006192449487634432000087081320919859200000870813209198592000006349679650406400000034016140984320000000132875550720000000003690987520000000000692060160000000000786432000000000004096000000000000390877006486250192896
11368934881474191032322997595911977802137601124098466991675801600257605898685592371200040250921669623808000004528228687832678400000377352390652723200000023584524415795200000001105524581990400000000383862702080000000000959656755200000000001635778560000000000017039360000000000008192000000000000020325604337285010030592
121180591620717411303424103301766812773489049604196634276768922992640010491585691922307481600018032412907991465984000022540516134989332480000021131733876552499200000015094095626108928000000082545835455283200000000343940981063680000000001074815565824000000000024427626496000000000003816816640000000000003670016000000000000016384000000000000001056931425538820521590784
1337778931862957161709568354177486215223391027200154952650219160233574400041966342767689229926400007868689268941730611200000108194477447948795904000001127025806749466624000000090564573756653568000000005660285859790848000000000275152784850944000000000010318229431910400000000002931315179520000000000006106906624000000000000088080384000000000000007864320000000000000003276800000000000000054960434128018667122720768
1412089258196146291747061761208925819614629174706176056668397794435742564352000165282826900437582479360000335730742141513839411200000503596113212270759116800000577037213055726911488000000515211797371184742400000000362258295026614272000000000201254608348119040000000000880488911523020800000000003001666743828480000000000078168404787200000000000001503238553600000000000000201326592000000000000000167772160000000000000006553600000000000000002857942574656970690381479936
Technical details
Node id
n-6590a4d05a99a1a1 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-4b28493176e02e0b

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 Example
Triangle begins:
  0:              1024,               1280,                400;
  1:             32768,              61440,              38400,                8000;
  2:           1048576,            2621440,            2457600,             1024000,              160000;
  3:          33554432,          104857600,          131072000,            81920000,            25600000,             3200000;
  4:        1073741824,         4026531840,         6291456000,          5242880000,          2457600000,           614400000,           64000000;
  5:       34359738368,       150323855360,       281857228800,        293601280000,        183500800000,         68812800000,        14336000000,         1280000000;
  6:     1099511627776,      5497558138880,     12025908428800,      15032385536000,      11744051200000,       5872025600000,      1835008000000,       327680000000,        25600000000;
  7:    35184372088832,    197912092999680,    494780232499200,     721554505728000,     676457349120000,     422785843200000,    176160768000000,     47185920000000,      7372800000000,      512000000000;
  8:  1125899906842624,   7036874417766400,  19791209299968000,   32985348833280000,   36077725286400000,   27058293964800000,  14092861440000000,   5033164800000000,   1179648000000000,   163840000000000,   10240000000000;
  9: 36028797018963968, 247697979505377280, 774056185954304000, 1451355348664320000, 1814194185830400000, 1587419912601600000, 992137445376000000, 442918502400000000, 138412032000000000, 28835840000000000, 3604480000000000, 204800000000000;
%F Formula
The row polynomials satisfy P(n,x) = (32+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 1280*x + 400*x^2.
%t Mathematica
P[0, x_] := 1024 + 1280*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 20*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1024 + 1280*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 20*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

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.