Real-rooted atlas

A038333 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(21 + 19\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 441 + 798\,x + 361\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
04417983611600
192612513722743685964000
21944817038369552065761561303212560000
34084101184756953343221030248190136837052476099102400000
48576612146558751410531146151270423980862073415311988474470458814096000000
5180108854111406894093309615696814668808126542241597335229311528396915744507893871739163840000000
637822859361273765458232866923951068156871953050417741470880701284144558984580922538588150170452152169835630416553600000000
779428004658164677089507312340694667883649414665210876670627599290466067583041199436598119931044141911077283643209893414749322687697779262144000000000
816679880978201150913208850390614432350319445148243995632628023471965975166102548384877303748192140129637981099337754098548033703880854864567764416533590613106625780110485760000000000
935027750054222134860951244440091577043032486575542805453738921335774574877180481309811281777619429888768739892747222573675529919114702595198825824566578267901096296451416276305552031116490258898219419430400000000000
1073558275113866417986327012362638839741484418661702611985527046897973802439910863118516095353206143994299472837282870754953833242891324670792338088163497526026947689565745036920889018017845081449955590629355545242351188221331491906616116777216000000000000
11154472377739119461181688939531250032798631138602678589183272048883803146847474010629514594988215120531596638054695093145403195944319949636131555272521051383004892696492107134384674487098769850675478516239024119459587446400703192558093716260423497290506195342052983462257059671088640000000000000
123243919932521508681410896524786057766262416462895765625434918745294289437501573962175912507729092653521393736549017645337303853435674509537581491235525321445884158086168437421281132495848488326384140766721971813581193568272780279677281392689128706931841876888225410170441070911236357713790357534679900668578288412126843545600000000000000
136812231858295168230192451718076862997408558552754782013231692052295639750977344163164562310221812242198714465124027012940558281250697187024860783381535219305217559531931688724642865196839576509623131819735138516739025290352021295751948011851576196687033092426888567088254773593263668067896312445551947263399219813116995251687106021608498115151811270298747982991073741824000000000000000
141430568690241985328321207091848492173114195041405266114768317560609205933345817910674144794401744686210742781564005020378846148618432568183947262840353223216195833686488122222525449712386666960826726221340417153642887066488034732139368970221604210908866368826701447368207811086911708330720779278341481177032902454262021809350071261906910344042283433811630227683320510085868203793222846428844141356762116768142949672960000000000000000
Technical details
Node id
n-f8553e5f84e9ff5e · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-7745de9e47db299c

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:             441,              798,               361;
  1:            9261,            25137,             22743,              6859;
  2:          194481,           703836,            955206,            576156,            130321;
  3:         4084101,         18475695,          33432210,          30248190,          13683705,           2476099;
  4:        85766121,        465587514,        1053114615,        1270423980,         862073415,         311988474,          47045881;
  5:      1801088541,      11406894093,       30961569681,       46688081265,       42241597335,       22931152839,        6915744507,         893871739;
  6:     37822859361,     273765458232,      866923951068,     1568719530504,     1774147088070,     1284144558984,      580922538588,      150170452152,       16983563041;
  7:    794280046581,    6467708950731,    23406946678836,    49414665210876,    67062759929046,    60675830411994,    36598119931044,    14191107728364,     3209893414749,     322687697779;
  8:  16679880978201,  150913208850390,   614432350319445,  1482439956326280,  2347196597516610,  2548384877303748,  1921401296379810,   993377540985480,   337038808548645,   67764416533590,    6131066257801;
  9: 350277500542221, 3486095124444009, 15770430324865755, 42805453738921335, 77457487718048130, 98112817776194298, 88768739892747222, 57367552991911470, 25951988258245665, 7826790109629645, 1416276305552031, 116490258898219;
%F Formula
The row polynomials satisfy P(n,x) = (21+19*x)*P(n-1,x) for n >= 1, with P(0,x) = 441 + 798*x + 361*x^2.
%t Mathematica
P[0, x_] := 441 + 798*x + 361*x^2;
P[n_, x_] := P[n, x] = Expand[(21 + 19*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] = 441 + 798*x + 361*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 19*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.