Real-rooted atlas

A038336 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(44 + 40\,x + 22\,n + 20\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 880 + 800\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
08808001680
1290405280024000105840
2851840232320021120006400005927040
323425600851840001161600007040000016000000311169600
4618435840281107200051110400004646400000211200000038400000015682947840
51587318656086581017600196775040000238515200000162624000000591360000008960000000768464444160
6399097262080253970984960069264814080001049466880000095406080000005203968000000157696000000020480000000036886293319680
79877657236480718375071744002285738864640004155888844800004722600960000003434618880000001561190400000004055040000000046080000000001742877359354880
824145384355840019755314472960007183750717440000152382590976000002077944422400000018890403840000000114487296000000004460544000000000101376000000000010240000000000081334276769894400
95843183014113280531198455828480002173084592025600005268083859456000008381042503680000009142955458560000006926481408000000003598172160000000001226649600000000002478080000000000022528000000000003757643586769121280
1014023639233871872014023639233871872006374381469941760000173846767362048000003160850315673600000040229004017664000000365718218342400000002374793625600000000010794516480000000000327106560000000000059473920000000000049152000000000000172168397066512465920
113342300684072796160364614620080668672001823073100403343360005524463940616192000001130003987853312000000164364216415027200000017432568407654400000001358381953843200000000771807928320000000000311841587200000000000850477056000000000001405747200000000000010649600000000000007833662066526317199360
12791868162072631705609358441915403829248005104604681129361408000170153489370978713600003867124758431334400000063280223319785472000000767033009936793600000006973027363061760000000047543368384512000000000240118022144000000000008731564441600000000000216485068800000000000032800768000000000000022937600000000000000354322561162882654863360
1318665463820283461632002375604486217895116800014037662873105743872000051046046811293614080000012761511702823403520000002320274855058800640000000316401116598927360000000032872843283005440000000002614885261148160000000000158477894615040000000000072035406643200000000000023813357568000000000000054121267200000000000000756940800000000000000049152000000000000000015944515252329719468851200
144380162176493185662976059729484224907077222400038009671779486321868800001497350706464612679680000040836837449034891264000000816736748980697825280000001237479922698027008000000001446405104452239360000000001314913731320217600000000009297369817415680000000000050712926276812800000000000209557546598400000000000006350228684800000000000000133221580800000000000000017301504000000000000000010485760000000000000000714314283304371432204533760
Technical details
Node id
n-20b892927bf76ac2 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c4264000c5d1dd22

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:              880,               800;
  1:            29040,             52800,              24000;
  2:           851840,           2323200,            2112000,             640000;
  3:         23425600,          85184000,          116160000,           70400000,           16000000;
  4:        618435840,        2811072000,         5111040000,         4646400000,         2112000000,          384000000;
  5:      15873186560,       86581017600,       196775040000,       238515200000,       162624000000,        59136000000,         8960000000;
  6:     399097262080,     2539709849600,      6926481408000,     10494668800000,      9540608000000,      5203968000000,      1576960000000,       204800000000;
  7:    9877657236480,    71837507174400,    228573886464000,    415588884480000,    472260096000000,    343461888000000,    156119040000000,     40550400000000,      4608000000000;
  8:  241453843558400,  1975531447296000,   7183750717440000,  15238259097600000,  20779444224000000,  18890403840000000,  11448729600000000,   4460544000000000,   1013760000000000,   102400000000000;
  9: 5843183014113280, 53119845582848000, 217308459202560000, 526808385945600000, 838104250368000000, 914295545856000000, 692648140800000000, 359817216000000000, 122664960000000000, 24780800000000000, 2252800000000000;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (22*n+44+(20*n+40)*x)*P(n-1,x) for n >= 1, with P(0,x) = 880 + 800*x.
%t Mathematica
P[0, x_] := 880 + 800*x;
P[n_, x_] := P[n, x] = Expand[Together[((22*n + 44 + (20*n + 40)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 880 + 800*x; for(n = 1, N, my(i = n + 1); v[i] = ((22*n + 44 + (20*n + 40)*x)*v[i - 1])/(n + 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.