A253284 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2\,n - n^{2}\right)\,P_{n-2}(x) + \left(-3 - 4\,x + 6\,n + 8\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	4														8
2	36	84	48													168
3	528	1968	2400	960												5856
4	10800	55440	105120	87360	26880											285600
5	283680	1856160	4798080	6128640	3870720	967680										17904960
6	9102240	72404640	237484800	411264000	396748800	202245120	42577920									1371827520
7	345058560	3232131840	12859015680	28174003200	36723456000	28484628480	12177285120	2214051840								124209630720
8	15090727680	162674023680	761184345600	2019630412800	3323951769600	3475507875840	2255011799040	830269440000	132843110400							12976163504640
9	747888422400	9118920384000	49072117248000	152986724352000	304542830592000	401485090406400	350572968345600	195545058508800	63233320550400	9033331507200						1536338250316800
10	41422381862400	563604060096000	3429253884672000	12288153765888000	28720016252928000	45751864648089600	50315125122662400	37723192374067200	18455096269209600	5320632257740800	686533194547200					203294894211763200
11	2535569103513600	38083762744473600	258527596174848000	1047065327852544000	2811416200040448000	5255101814125363200	6978302789301043200	6583672669077504000	4325159125647360000	1884533619032064000	490184700906700800	57668788341964800				29732251962347827200
12	169983582318950400	2793419499680409600	20930599842395443200	94557468018004992000	286871326014873600000	615762091820600524800	958934444346585907200	1091750487697155686400	901919253672493056000	527308983401840640000	207117453330166579200	49076138879012044800	5305528527460761600			4762497178632588595200
13	12386182292118835200	221058842635466035200	1812167491894695936000	9035139743707004928000	30569063736310235136000	74117809928612472422400	132498395209183946342400	176833265820267184128000	176200495040184975360000	129470610814865768448000	68194610927716899225600	24384209112209660313600	5305528527460761600000	530552852746076160000		829185294230087264870400
14	974723523832041984000	18774179465075274240000	167143581074578268160000	911674753737869721600000	3403694872190986813440000	9201553456313254772736000	18575992497583820242944000	28449907132100385964032000	33218710372226860056576000	29428185031132595355648000	19471793720991103844352000	9331894126950733578240000	3062351066050351595520000	615971862038194421760000	57299708096576225280000	155915921083476218176512000

Technical details

Node id: n-3e752e21877e7b02 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-8419344160e8e665

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:            1;
  1:            4,             4;
  2:           36,            84,             48;
  3:          528,          1968,           2400,             960;
  4:        10800,         55440,         105120,           87360,           26880;
  5:       283680,       1856160,        4798080,         6128640,         3870720,          967680;
  6:      9102240,      72404640,      237484800,       411264000,       396748800,       202245120,        42577920;
  7:    345058560,    3232131840,    12859015680,     28174003200,     36723456000,     28484628480,     12177285120,      2214051840;
  8:  15090727680,  162674023680,   761184345600,   2019630412800,   3323951769600,   3475507875840,   2255011799040,    830269440000,   132843110400;
  9: 747888422400, 9118920384000, 49072117248000, 152986724352000, 304542830592000, 401485090406400, 350572968345600, 195545058508800, 63233320550400, 9033331507200;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-2*n)*P(n-2,x) + (6*n-3+(8*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n)*P[n - 2, x] + (6*n - 3 + (8*n - 4)*x)*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; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n)*v[i - 2] + (6*n - 3 + (8*n - 4)*x)*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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] 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

Related ?

Same fingerprint ?

A088617 → basis A013609

References

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