Real-rooted atlas

A034870 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(25 + 30\,x + 9\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
12530964
262515001350540814096
315625562508437567500303757290729262144
43906251875000393750047250003543750170100051030087480656116777216
597656255859375015820312525312500026578125019136250095681250328050007381125984150590491073741824
6244140625175781250058007812501160156250015662109375150356250001052493750054128250002029809375541282500974308501062882053144168719476736
7610351562551269531250199951171875479882812500791806640625950167968750855151171875586389375000307854421875123141768750369425306258060188500120902827511160261047829694398046511104
815258789062514648437500006591796875000184570312500003599121093750051827343750000570100781250004886578125000032984402343750175916812500007388506125000241805655000060451413750011160261000014348907000114791256043046721281474976710656
9381469726562541198730468750210113525390625672363281250000151281738281250025415332031250003303993164062500339839296875000028036741992187501869116132812500100932271171875044043172875000015415110506250042687998325000914742821250014635885140001646537078251162261467038742048918014398509481984
109536743164062511444091796875006523132324218750234832763671875005988235473632812511497412109375000017246118164062500020695341796875000020177958251953125016142366601562500010653961957031250058112519765625000261506338945312509655618668750000289668560062500069520454415000013035085202812518402473227500184024732275011622614670034867844011152921504606846976
112384185791015625314712524414062501982688903808593757930755615234375002260265350341796875488217315673828125082996943664550781251138243798828125000012805242736816406250119515598876953125009322216712402343750610181457539062500033559980164648437501548922161445312500597441405128906250191181249641250000501850780308281251062742828888125017712380481468752237353323975002013617991577511506388523303138105960973786976294838206464
1259604644775390625858306884765625000592231750488281250026058197021484375000820833206176757812501969999694824218750003742999420166015625005774913391113281250007363014573669433593757853882211914062500007068493990722656250005397759047460937500003508543380849609375001943193257085937500009160768211976562500036643072847906250000123670370861683593753491869294918125000814769502147562500154377379354275000231566069031412502646469360359000216529311302100112971814592402824295364814722366482869645213696
131490116119384765625232458114624023437501743435859680175781258368492126464843750002887129783630371093750762202262878417968750016006247520446777343750274392814636230468750003910097608566284179687546921171302795410156250478595947288513183593754176837358154296875000031326280186157226562500202415964279785156250001127746086701660156250054131812161679687500002232937251669287109375788095500589160156250236428650176748046875597293432025468750001254316207253484375021502563552916875002932167757215937503059653311877500022947399839081251101475192275902541865828329302231454903657293676544
14372529029846191406256258487701416015625005069375038146972656250263607501983642578125009885281324386596679687528469610214233398437500065480103492736816406250012347676658630371093750001944759073734283447265625259301209831237792968750029560337920761108398437502902287723129272460937500246694456465988159179687518217436785180664062500001171120936190185546875000655827724266503906250000319716015579920654296875135409136010319335937500496500165371170898437501567895259066855468750042333171994805097656259676153598812593750001847265687046040625002891372379724237500036142154746552968753469646855669085002402063207770905010675836478981802287679245496119342813113834066795298816
Technical details
Node id
n-ff1fa688e6937f48 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-eb1af6eb566f4016

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:            25,             30,               9;
  2:           625,           1500,            1350,             540,               81;
  3:         15625,          56250,           84375,           67500,            30375,             7290,              729;
  4:        390625,        1875000,         3937500,         4725000,          3543750,          1701000,           510300,            87480,             6561;
  5:       9765625,       58593750,       158203125,       253125000,        265781250,        191362500,         95681250,         32805000,          7381125,           984150,            59049;
  6:     244140625,     1757812500,      5800781250,     11601562500,      15662109375,      15035625000,      10524937500,       5412825000,       2029809375,        541282500,         97430850,        10628820,          531441;
  7:    6103515625,    51269531250,    199951171875,    479882812500,     791806640625,     950167968750,     855151171875,     586389375000,     307854421875,     123141768750,      36942530625,      8060188500,      1209028275,      111602610,       4782969;
  8:  152587890625,  1464843750000,   6591796875000,  18457031250000,   35991210937500,   51827343750000,   57010078125000,   48865781250000,   32984402343750,   17591681250000,    7388506125000,   2418056550000,    604514137500,   111602610000,   14348907000,    1147912560,     43046721;
  9: 3814697265625, 41198730468750, 210113525390625, 672363281250000, 1512817382812500, 2541533203125000, 3303993164062500, 3398392968750000, 2803674199218750, 1869116132812500, 1009322711718750, 440431728750000, 154151105062500, 42687998325000, 9147428212500, 1463588514000, 164653707825, 11622614670, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (25+30*x+9*x^2)*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[(25 + 30*x + 9*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (25 + 30*x + 9*x^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.