A139548 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(50 + 120\,x + 72\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	50	120	72																											242
2	2500	12000	21600	17280	5184																									58564
3	125000	900000	2700000	4320000	3888000	1866240	373248																							14172488
4	6250000	60000000	252000000	604800000	907200000	870912000	522547200	179159040	26873856																					3429742096
5	312500000	3750000000	20250000000	64800000000	136080000000	195955200000	195955200000	134369280000	60466176000	16124313600	1934917632																			829997587232
6	15625000000	225000000000	1485000000000	5940000000000	16038000000000	30792960000000	43110144000000	44341862400000	33256396800000	17736744960000	6385228185600	1393140695040	139314069504																	200859416110144
7	781250000000	13125000000000	102375000000000	491400000000000	1621620000000000	3891888000000000	7005398400000000	9607403520000000	10087773696000000	8070218956800000	4842131374080000	2112930054144000	633879016243200	117023818383360	10030613004288															48607978698654848
8	39062500000000	750000000000000	6750000000000000	37800000000000000	147420000000000000	424569600000000000	934053120000000000	1601233920000000000	2161665792000000000	2305776844800000000	1936852549632000000	1267758032486400000	633879016243200000	234047636766720000	60183678025728000	9629388484116480	722204136308736													11763130845074473216
9	1953125000000000	42187500000000000	430312500000000000	2754000000000000000	12393000000000000000	41640480000000000000	108265248000000000000	222717081600000000000	367483184640000000000	489977579520000000000	529175785881600000000	461826140405760000000	323278298284032000000	179046442126540800000	76734189482803200000	24554940634497024000	5524861642761830400	779980467213434880	51998697814228992											2846677664508022518272
10	97656250000000000	2343750000000000000	26718750000000000000	192375000000000000000	981112500000000000000	3767472000000000000000	11302416000000000000000	27125798400000000000000	52895306880000000000000	84632491008000000000000	111714888130560000000000	121870787051520000000000	109683708346368000000000	80997200009625600000000	48598320005775360000000	23327193602772172800000	8747697601039564800000	2469938146175877120000	493987629235175424000	62398437377074790400	3743906242624487424									688895994810941449421824
11	4882812500000000000	128906250000000000000	1624218750000000000000	12993750000000000000000	74064375000000000000000	319958100000000000000000	1087857540000000000000000	2983837824000000000000000	6713635104000000000000000	12532118860800000000000000	19550105422848000000000000	25592865280819200000000000	28152151808901120000000000	25986601669754880000000000	20046807002382336000000000	12829956481524695040000000	6735727152800464896000000	2852778558833138073600000	950926186277712691200000	240233983901737943040000	43242117102312829747200	4941956240264323399680	269561249468963094528							166712830744247830760081408
12	244140625000000000000	7031250000000000000000	97031250000000000000000	853875000000000000000000	5379412500000000000000000	25821180000000000000000000	98120484000000000000000000	302771779200000000000000000	772068036960000000000000000	1647078478848000000000000000	2964741261926400000000000000	4527968472760320000000000000	5886359014588416000000000000	6520274600774860800000000000	6147687480730583040000000000	4918149984584466432000000000	3319751239594514841600000000	1874683052947490734080000000	874852091375495675904000000	331522897784398361395200000	99456869335319508418560000	22732998705215887638528000	3719945242671690704486400	388168199235306856120320	19408409961765342806016					40344505040107975043939700736
13	12207031250000000000000	380859375000000000000000	5712890625000000000000000	54843750000000000000000000	378421875000000000000000000	1998067500000000000000000000	8391883500000000000000000000	28772172000000000000000000000	82000690200000000000000000000	196801656480000000000000000000	401475379219200000000000000000	700757025546240000000000000000	1051135538319360000000000000000	1358390541828096000000000000000	1513635175179878400000000000000	1453089768172683264000000000000	1198799058742463692800000000000	846211100288797900800000000000	507726660173278740480000000000	256535575666498732032000000000	107744941779929467453440000000	36941122895975817412608000000	10074851698902495657984000000	2102577745857912137318400000	315386661878686820597760000	30277119540353934777384960	1397405517247104682033152			9763370219706129960633407578112
14	610351562500000000000000	20507812500000000000000000	332226562500000000000000000	3455156250000000000000000000	25913671875000000000000000000	149262750000000000000000000000	686608650000000000000000000000	2589495480000000000000000000000	8156910762000000000000000000000	21751762032000000000000000000000	49594017432960000000000000000000	97384616050176000000000000000000	165553847285299200000000000000000	244510297529057280000000000000000	314370382537359360000000000000000	352094828441842483200000000000000	343292457730796421120000000000000	290788905371968733184000000000000	213245197272777071001600000000000	134681177224911834316800000000000	72727835701452390531072000000000	33247010606378235671347200000000	12694313140617144529059840000000	3973871939671453939531776000000	993467984917863484882944000000	190745853104229789097525248000	26410964275970278490426572800	2347641268975135865815695360	100613197241791537106386944	2362735593168883450473284633903104

Technical details

Node id: n-3b4b7a0ddd26d13e · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-0d4cb94e85dd3af7

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:               50,               120,                 72;
  2:             2500,             12000,              21600,               17280,                 5184;
  3:           125000,            900000,            2700000,             4320000,              3888000,              1866240,                373248;
  4:          6250000,          60000000,          252000000,           604800000,            907200000,            870912000,             522547200,             179159040,              26873856;
  5:        312500000,        3750000000,        20250000000,         64800000000,         136080000000,         195955200000,          195955200000,          134369280000,           60466176000,           16124313600,            1934917632;
  6:      15625000000,      225000000000,      1485000000000,       5940000000000,       16038000000000,       30792960000000,        43110144000000,        44341862400000,        33256396800000,        17736744960000,         6385228185600,         1393140695040,          139314069504;
  7:     781250000000,    13125000000000,    102375000000000,     491400000000000,     1621620000000000,     3891888000000000,      7005398400000000,      9607403520000000,     10087773696000000,      8070218956800000,      4842131374080000,      2112930054144000,       633879016243200,       117023818383360,       10030613004288;
  8:   39062500000000,   750000000000000,   6750000000000000,   37800000000000000,   147420000000000000,   424569600000000000,    934053120000000000,   1601233920000000000,   2161665792000000000,   2305776844800000000,   1936852549632000000,   1267758032486400000,    633879016243200000,    234047636766720000,    60183678025728000,     9629388484116480,     722204136308736;
  9: 1953125000000000, 42187500000000000, 430312500000000000, 2754000000000000000, 12393000000000000000, 41640480000000000000, 108265248000000000000, 222717081600000000000, 367483184640000000000, 489977579520000000000, 529175785881600000000, 461826140405760000000, 323278298284032000000, 179046442126540800000, 76734189482803200000, 24554940634497024000, 5524861642761830400, 779980467213434880, 51998697814228992;

%F Formula

The row polynomials satisfy P(n,x) = (50+120*x+72*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[(50 + 120*x + 72*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] = (50 + 120*x + 72*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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from basis_map_A007318 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
    
    Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

A034870 → basis A013610 → basis A013609

References

[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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.