Real-rooted atlas

A038314 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(23 + 12\,x + 12\,n\right)\,P_{n}(x) = \left(770 + 1104\,x + 288\,x^{2} + 264\,n + 288\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1540 + 2208\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0154022085764324
1454961108807948813824249688
2125646443676165322240254361633177613821712
333264352150775680262056960212889600763084807962624743257696
4855502912479006668810855848960125787340807664025600219768422419110297639132965440
5215421817601437244892164023656017926079275417605283068313602575112601606153515827245864714242027499535744
653379258803241360988979201379755096473625751398514688291805220044802028698232422482403603251201687821484032110075314176103724602417408
713060457901568115299199014912446698680975360993423669460992139057551979315212605985505935367303313636720642542739757465604557118006886426418075402245252474404767232
831630453506150431345098963763201383590388178944035735894478028800596054201676595206674762495007129650423942023741440250399324687564807628219272396800121523146850304063403380965376263746386523350016
97596137918347264835043972562370564137553063216742401217559541597470720235856903554990080031471661848524226562936895497803137024190170295632396288082631777146896384022376109865697280032082110768480256152168114316902413150431719660738560
1018113867343751168021876877204840120321202463320489813606439720509406880727040876642869950178918401358535764476742860801510639768729162874881208322719096147804166846130642766266368026442168687006842880644431964132081664083996799102930124836520347436056576651752364325379313664
1142935708787704381445651526611250364416034127928439550587699212505618533094061506563098199733736696709120547025150848911645081670643859752790628761606733137254907125956608471245860447497643622423733252894923056742408249956630346134978561827843389174631628802183916776676183244887648833846535782432134903999132774064128
12101246000722143625216144263981526686721638494945647069006122188803822327985229665822310410504719567799011665510420819902210710601885286430633408447539052124569633909052681339501805568028279176470609929017753617593178790039912028569679743729726941470654464251998675254209213890565117961489688968560640564458489971690438656210357201231685877761577303165007712586186752
1323767357264494274478083644856025997170507776025967516674803609894912011393477648280734662656003440095186706699240079360756339808881528839916748812491941326426361131171840157543243444486553783500801525907370660277581250560011311670588243971607101440633354436441436833028505626097947910635390396006407559960257626276416716801417281643298483601408001451464688498632556544050485728295604610662477114434318544608009486336
145557330761425729309900891266651895658013995827269981235699145673749299203323842134374862066548736010937738542349505276149760026419931033907450163809484848405747768417845754671923268527220990681752490999808075620756853353545816080384065105381148171843466690560043436815058856850971269529622109827599410158534813286483513433314033249267220480223309595302191549539942403887401078761555021004800371574960255649934475264121165747909451065589763756954460149784104501575680
Technical details
Node id
n-610f511ce0971e21 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-29d09fbfedb73813

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:             1540,              2208,                576;
  1:            45496,            110880,              79488,               13824;
  2:          1256464,           4367616,            5322240,             2543616,              331776;
  3:         33264352,         150775680,          262056960,           212889600,            76308480,             7962624;
  4:        855502912,        4790066688,        10855848960,         12578734080,          7664025600,          2197684224,           191102976;
  5:      21542181760,      143724489216,       402365601792,        607927541760,        528306831360,        257511260160,         61535158272,          4586471424;
  6:     533792588032,     4136098897920,     13797550964736,      25751398514688,      29180522004480,      20286982324224,       8240360325120,       1687821484032,       110075314176;
  7:   13060457901568,   115299199014912,    446698680975360,     993423669460992,    1390575519793152,    1260598550593536,     730331363672064,     254273975746560,     45571180068864,      2641807540224;
  8:  316304535061504,  3134509896376320,  13835903881789440,   35735894478028800,   59605420167659520,   66747624950071296,   50423942023741440,   25039932468756480,   7628219272396800,   1215231468503040,    63403380965376;
  9: 7596137918347264, 83504397256237056, 413755306321674240, 1217559541597470720, 2358569035549900800, 3147166184852422656, 2936895497803137024, 1901702956323962880, 826317771468963840, 223761098656972800, 32082110768480256, 1521681143169024;
%F Formula
The row polynomials satisfy (12*n+23+12*x)*P(n,x) = (264*n+770+(288*n+1104)*x+288*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1540 + 2208*x + 576*x^2.
%t Mathematica
P[0, x_] := 1540 + 2208*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((264*n + 770 + (288*n + 1104)*x + 288*x^2)*P[n - 1, x])/(12*n + 23 + 12*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1540 + 2208*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((264*n + 770 + (288*n + 1104)*x + 288*x^2)*v[i - 1])/(12*n + 23 + 12*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. [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.