A241171 → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard 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 | Pn(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 |
Compute more
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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 ExampleTriangle 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 FormulaThe 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 MathematicaP[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/GPP(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]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
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.