Real-rooted atlas

A034870 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 24\,x + 9\,x^{2} + 36\,n x - 36\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(16 + 96\,x + 216\,x^{2} + 216\,x^{3} + 81\,x^{4} + 144\,n x + 576\,n x^{2} + 756\,n x^{3} + 324\,n x^{4} + 144\,n^{2} x^{2} + 432\,n^{2} x^{3} + 324\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
144881133
2163841944345620257825
3642304194406912012150010497635721353125
425612288145152774144226800039191044000752223948853144113890625
51024614409331206635520272160007054387212002256013436928095659380393660007144929501953125
64096294912547430448660480256608000886837248211239705635473489924209012720346420800018862612566122200328981352917119140625
716384137625630191616322043904207567360089669099522746116172861487382528102138708672126097171200114433182864742826972163269212455687496446241076168025559814453125
8655366291456159252480198180864015095808000782566686722929190584328198317670401750949291520288222105600036618618516483565569466368261536996448013999431398405165606520001175462461441244050236917730712890625
926214428311552812187648115511132161015234560006140138618882716158541824912249166233623812910364672489977579520008003783761459210391088159129610670709455078485676520158208526891865040002397943421337676135874498281506290861232139858796529547637939453125
1010485761258291204034396160645503385606429818880004444290809856226844010086408888581619712027420927086592067705992806400013517501463797762193674166927360289633542352128031006740628684802669585449536000182243700021657696438774364488038159368484544010629268536204018596183472000153767192084116574859619140625
11419430455364812819619905536348798320640388310630400030195034619904174669887766528782195182534656278427875033088080205560709120001892450204931686436853726004379648594714206963036167958396761362432088096319834688000801872280095293445940628500852460835259256479718656163690735457541605727624509376000142080885485708422276368181108816600580533161493335723876953125
12167772162415919104937678602241833682599936225628913664001949433814056961260363111727104634958442292838425615364503044096843304181170176002295895633235804165216219680619888649947946734654423041597467277189840896216129637994434560024590749922922332162342304951764684544185362948350520934412047638129675061766323297458351104002614288292937034568197703490648038418327040908609744260287060820889617651846030062514479160308837890625
13671088641046898278444166021120094220845056001269773107200001206792361082880862353708023808048271694443315200217646887934361600806099584942080000248722026933878784064581767474341478401421135247807774720026624454619830681600425709893019340800005812359072690733056067666587495424220160669366202376881152005593546274491992960039144222361221120000226571652054542995201065701453784244992039708588635321112001127910596890521600229473998390812500297838891991403361853020188851841419914722442626953125
1426843545645097156608205473919795247487305908224695614832640000721213458481152056444969979740160347556199991869440173201102398291968071276173826457600002457958854405390336071799729721473761280179063041223779614720383392146525561815040707333360709058560000112670345101389594624015501800044406275891201840148541806989639680187943154822930963456016440573391712870400001223486921094532174080767305046724656394240400262573444036808960170540082249846865920578274475944847500001501108015636672934428017665255439835923347942319808692481923938245462220112053549289703369140625
Technical details
Node id
n-a43d8d44a6eb5699 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d115b0993943a210

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 Example
Triangle begins:
  0:      1;
  1:      4,       48,        81;
  2:     16,      384,      1944,        3456,         2025;
  3:     64,     2304,     19440,       69120,       121500,       104976,         35721;
  4:    256,    12288,    145152,      774144,      2268000,      3919104,       4000752,       2239488,         531441;
  5:   1024,    61440,    933120,     6635520,     27216000,     70543872,     120022560,     134369280,       95659380,       39366000,        7144929;
  6:   4096,   294912,   5474304,    48660480,    256608000,    886837248,    2112397056,    3547348992,     4209012720,     3464208000,     1886261256,       612220032,        89813529;
  7:  16384,  1376256,  30191616,   322043904,   2075673600,   8966909952,   27461161728,   61487382528,   102138708672,   126097171200,   114433182864,     74282697216,     32692124556,     8749644624,     1076168025;
  8:  65536,  6291456, 159252480,  1981808640,  15095808000,  78256668672,  292919058432,  819831767040,  1750949291520,  2882221056000,  3661861851648,   3565569466368,   2615369964480,  1399943139840,   516560652000,   117546246144,   12440502369;
  9: 262144, 28311552, 812187648, 11551113216, 101523456000, 614013861888, 2716158541824, 9122491662336, 23812910364672, 48997757952000, 80037837614592, 103910881591296, 106707094550784, 85676520158208, 52689186504000, 23979434213376, 7613587449828, 1506290861232, 139858796529;
%F Formula
The row polynomials satisfy (4+(36*n-24)*x+(36*n^2-36*n+9)*x^2)*P(n,x) = (16+(144*n+96)*x+(144*n^2+576*n+216)*x^2+(432*n^2+756*n+216)*x^3+(324*n^2+324*n+81)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16 + (144*n + 96)*x + (144*n^2 + 576*n + 216)*x^2 + (432*n^2 + 756*n + 216)*x^3 + (324*n^2 + 324*n + 81)*x^4)*P[n - 1, x])/(4 + (36*n - 24)*x + (36*n^2 - 36*n + 9)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16 + (144*n + 96)*x + (144*n^2 + 576*n + 216)*x^2 + (432*n^2 + 756*n + 216)*x^3 + (324*n^2 + 324*n + 81)*x^4)*v[i - 1])/(4 + (36*n - 24)*x + (36*n^2 - 36*n + 9)*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.