Real-rooted atlas

A034870 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + x + 2\,n\right)\,P_{n}(x) = \left(4 + 12\,x + 12\,x^{2} + 4\,x^{3} + 8\,n + 16\,n x + 8\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
11216432
28025628812816768
344823044800512028807686416384
423041638450176860168960057344215044096256327680
5112641024004147209830401505280154828810752004915201382402048010246291456
6532485898242973696901120018247680259522562649292819464192101376003604480811008983044096117440512
72457603211264193822727156531218040422432800768044281036844983910434440806419680460882001920238551044472832458752163842147483648
8111411216777216117964800513802240155058176034351349765772935168749731840075910348805997854720367368601617175674885963776001468006402359296020971526553638654705664
9498073684934656681836544342255206412032409600314446970886326373580810010964787212617986867212745441280010323807436866739765248340650885121347629875240108032008556380161203240969437184262144687194767360
1022020096419430400378535936021516779520863659622402601139568646096420864001137998561280171715854336021134258995202131037782016176118824960011888020684806502848921602844996403209754273382425401753600478150656059768832041943040104857612094627905536
1196468992203004313620346568704129184563200582945341440198815042764853201432739841144501410201620118188851200292086297395203525898875699235505555111936298345289482242086330695680012070913310720572250705100821906472304646627168092161534066688002583691264029066526721845493764194304211106232532992
124194304009663676416106501767168747055874048374376864153614261975777280429047771299841045199081963522097656490885123509804324618244935662331494405862968587714565897867210260485025401646612483619485709762562193627702886401110524024586244645329253171215807023153152427859273318489137348608013582834073613891534848805306368167772163659174697238528
131811939328453655920645452595200004187593113600230753828864009711726362624032445994893312088288421478400019920075096064003774330018201600605989652922368082959011309158409721759137792000977161431285760084255245860864006221925848186880392110951890944020968500101120009435825045504003531536859136001081533163110402648652644352050163875840006979321856006543114240034896609286710886463050394783186944
1477846282242104533975042739652263936228642584002561374053990400006331640787763202325998594949120699244298108928017520837242388480370811370209280006693145232277504010375976340946944013882672048766976016081285269159936016153076721254400014071124610514944010616160978468864069173175606312960387497881868697601854056851046400075089302467379202542706538577920707912615854080158291019694080274810798080003517578215424304405807104150323855362684354561080863910568919040
Technical details
Node id
n-ba46e27673af7615 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-c94e3f620e523ccf

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:      12,       16,         4;
  2:      80,      256,       288,        128,          16;
  3:     448,     2304,      4800,       5120,        2880,         768,          64;
  4:    2304,    16384,     50176,      86016,       89600,       57344,       21504,         4096,          256;
  5:   11264,   102400,    414720,     983040,     1505280,     1548288,     1075200,       491520,       138240,        20480,         1024;
  6:   53248,   589824,   2973696,    9011200,    18247680,    25952256,    26492928,     19464192,     10137600,      3604480,       811008,       98304,        4096;
  7:  245760,  3211264,  19382272,   71565312,   180404224,   328007680,   442810368,    449839104,    344408064,    196804608,     82001920,    23855104,     4472832,      458752,      16384;
  8: 1114112, 16777216, 117964800,  513802240,  1550581760,  3435134976,  5772935168,   7497318400,   7591034880,   5997854720,   3673686016,  1717567488,   596377600,   146800640,   23592960,   2097152,     65536;
  9: 4980736, 84934656, 681836544, 3422552064, 12032409600, 31444697088, 63263735808, 100109647872, 126179868672, 127454412800, 103238074368, 66739765248, 34065088512, 13476298752, 4010803200, 855638016, 120324096, 9437184, 262144;
%F Formula
The row polynomials satisfy (2*n-1+x)*P(n,x) = (8*n+4+(16*n+12)*x+(8*n+12)*x^2+4*x^3)*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[((8*n + 4 + (16*n + 12)*x + (8*n + 12)*x^2 + 4*x^3)*P[n - 1, x])/(2*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 4 + (16*n + 12)*x + (8*n + 12)*x^2 + 4*x^3)*v[i - 1])/(2*n - 1 + x)); 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.