A241171 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 + 13\,x + 47\,x^{2} + 79\,x^{3} + 70\,x^{4} + 32\,x^{5} + 6\,x^{6} - 3\,n x - 24\,n x^{2} - 63\,n x^{3} - 75\,n x^{4} - 42\,n x^{5} - 9\,n x^{6} + n^{2} x^{2} + 6\,n^{2} x^{3} + 12\,n^{2} x^{4} + 10\,n^{2} x^{5} + 3\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(3\,x + 28\,x^{2} + 94\,x^{3} + 156\,x^{4} + 139\,x^{5} + 64\,x^{6} + 12\,x^{7} - 2\,n x^{2} - 14\,n x^{3} - 36\,n x^{4} - 44\,n x^{5} - 26\,n x^{6} - 6\,n x^{7}\right)\,P_{n-1}'(x) + \left(x^{2} + 8\,x^{3} + 25\,x^{4} + 40\,x^{5} + 35\,x^{6} + 16\,x^{7} + 3\,x^{8}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	7														8
2	1	32	121													154
3	1	129	1515	3907												5552
4	1	514	14766	103594	202741											321616
5	1	2051	134914	1972486	9785429	15430207										27325088
6	1	8196	1212135	33595400	320622135	1226346036	1619195761									3200979664
7	1	32773	10859997	547861985	9056469395	63493007871	197305208143	224061282907								494474723072
8	1	131078	97379428	8784955586	239025234370	2752506013706	15176712147148	39681513963758	39531606447181							97390246272256
9	1	524295	874317564	140061381852	6105105409662	109191122431146	954855190814700	4327484068926684	9761297081388057	8661323866026007						23820397371219968
10	1	2097160	7857847845	2230444161120	153497988983010	4129765159004352	54013027191077010	378693515297651520	1454755512281108445	2884453333240827160	2307185279184885001					7083386168647642624
11	1	8388617	70666104415	35537618724735	3831920621932650	152135600948101722	2874414182234504574	29352964381674735390	171223761049326335685	570271648278026118925	1008505284912214435571	734307168916191403507				2516691244849530785792
12	1	33554442	635734892586	566763825905770	95396928088234995	5524004864871921492	147502206365287607484	2115493372145111426532	17624037421858706912295	87853509809430871983970	258101418121885003138266	411867922303844239590642	275199311597682485597221			1052914814802404260765696
13	1	134217739	5720406073350	9047239010246482	2373492576526837435	199038318330545534457	7404462620704407067908	145668227578115951552460	1669432611664439627575239	11706440133742133074335085	50883661291820480581022806	133669170036311392630504194	194309002966177267694253085	119956934012963778952439407		512347915163742179541659648
14	1	536870924	51478151732851	144526996646720104	59074063592488716641	7144493486829276985252	366800872361366497136523	9741434117988479998941552	149832414143295380233951083	1422570656409397347420883732	8594082738560572359583180001	33088344431942807012290157224	78581466720030847621171302451	104883094326613894061050492604	60172884133239122388455031841	286902390859642414913802102784

Technical details

Node id: n-5638f00ca3f62529 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-99a58a27f26df251

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: 1,      7;
  2: 1,     32,       121;
  3: 1,    129,      1515,         3907;
  4: 1,    514,     14766,       103594,        202741;
  5: 1,   2051,    134914,      1972486,       9785429,        15430207;
  6: 1,   8196,   1212135,     33595400,     320622135,      1226346036,      1619195761;
  7: 1,  32773,  10859997,    547861985,    9056469395,     63493007871,    197305208143,     224061282907;
  8: 1, 131078,  97379428,   8784955586,  239025234370,   2752506013706,  15176712147148,   39681513963758,   39531606447181;
  9: 1, 524295, 874317564, 140061381852, 6105105409662, 109191122431146, 954855190814700, 4327484068926684, 9761297081388057, 8661323866026007;

%F Formula

The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1-(3*n-13)*x+(n^2-24*n+47)*x^2+(6*n^2-63*n+79)*x^3+(12*n^2-75*n+70)*x^4+(10*n^2-42*n+32)*x^5+(3*n^2-9*n+6)*x^6)*P(n-1,x) + (3*x-(2*n-28)*x^2-(14*n-94)*x^3-(36*n-156)*x^4-(44*n-139)*x^5-(26*n-64)*x^6-(6*n-12)*x^7)*d/dx P(n-1,x) + (x^2+8*x^3+25*x^4+40*x^5+35*x^6+16*x^7+3*x^8)*d^2/dx^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[Together[((1 - (3*n - 13)*x + (n^2 - 24*n + 47)*x^2 + (6*n^2 - 63*n + 79)*x^3 + (12*n^2 - 75*n + 70)*x^4 + (10*n^2 - 42*n + 32)*x^5 + (3*n^2 - 9*n + 6)*x^6)*P[n - 1, x] + (3*x - (2*n - 28)*x^2 - (14*n - 94)*x^3 - (36*n - 156)*x^4 - (44*n - 139)*x^5 - (26*n - 64)*x^6 - (6*n - 12)*x^7)*D[P[n - 1, x], x] + (x^2 + 8*x^3 + 25*x^4 + 40*x^5 + 35*x^6 + 16*x^7 + 3*x^8)*D[P[n - 1, x], {x, 2}])/(1 + 3*x + 3*x^2 + x^3)]];
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] = ((1 - (3*n - 13)*x + (n^2 - 24*n + 47)*x^2 + (6*n^2 - 63*n + 79)*x^3 + (12*n^2 - 75*n + 70)*x^4 + (10*n^2 - 42*n + 32)*x^5 + (3*n^2 - 9*n + 6)*x^6)*v[i - 1] + (3*x - (2*n - 28)*x^2 - (14*n - 94)*x^3 - (36*n - 156)*x^4 - (44*n - 139)*x^5 - (26*n - 64)*x^6 - (6*n - 12)*x^7)*deriv(v[i - 1], x) + (x^2 + 8*x^3 + 25*x^4 + 40*x^5 + 35*x^6 + 16*x^7 + 3*x^8)*deriv(deriv(v[i - 1], x), x))/(1 + 3*x + 3*x^2 + x^3)); 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 [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.