A330798: Triangle read by rows, interpolating between the central binomial coefficients and the central coefficients of the Catalan triangle. T(n, k) for 0 <= k <= n.
A330798 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernel
Recurrence from A330798
\(\left(8\,n + 16\,n^{2}\right)\,P_{n}(x) = \left(-27 + 162\,x + 288\,x^{2} + 90\,n - 276\,n x - 576\,n x^{2} - 15\,n^{2} + 162\,n^{2} x + 288\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-5 + 33\,x + 246\,x^{2} + 288\,x^{3} + 42\,n + 60\,n x - 198\,n x^{2} - 288\,n x^{3} - n^{2} - 9\,n^{2} x\right)\,P_{n-1}'(x) + \left(11\,x + 9\,x^{2} + 7\,n x + 9\,n x^{2}\right)\,P_{n-1}''(x) + \left(-216 - 576\,x - 432\,x^{2} + 36\,n + 276\,n x + 612\,n x^{2} + 72\,n^{2} + 72\,n^{2} x - 216\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(24 + 144\,x + 264\,x^{2} + 144\,x^{3} - 16\,n - 108\,n x^{3}\right)\,P_{n-2}'(x) + \left(-4 - 4\,x + 2\,n\right)\,P_{n-2}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | |||||||
| 1 | 2 | 2 | 4 | ||||||
| 2 | 6 | 15 | 9 | 30 | |||||
| 3 | 20 | 84 | 112 | 48 | 264 | ||||
| 4 | 70 | 420 | 900 | 825 | 275 | 2490 | |||
| 5 | 252 | 1980 | 5940 | 8580 | 6006 | 1638 | 24396 | ||
| 6 | 924 | 9009 | 35035 | 70070 | 76440 | 43316 | 9996 | 244790 | |
| 7 | 3432 | 40040 | 192192 | 495040 | 742560 | 651168 | 310080 | 62016 | 2496528 |
All 15 rows b-files: A330798, A330801 (row sums) ?
Compute more
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Technical details
- Node id
n-eb78d61c7c9e8c34· revision 0 · JSON- Definition source
oeis_catalog:A330798- Fingerprint
f6-da93e8b7035ccffa
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 ?
Row sums ? A330801
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 2, 2; 2: 6, 15, 9; 3: 20, 84, 112, 48; 4: 70, 420, 900, 825, 275; 5: 252, 1980, 5940, 8580, 6006, 1638; 6: 924, 9009, 35035, 70070, 76440, 43316, 9996; 7: 3432, 40040, 192192, 495040, 742560, 651168, 310080, 62016; 8: 12870, 175032, 1002456, 3174444, 6104700, 7325640, 5372136, 2206413, 389367; 9: 48620, 755820, 5038800, 18992400, 44767800, 68643960, 68643960, 43263000, 15622750, 2466750;
%F FormulaThe row polynomials satisfy (16*n^2+8*n)*P(n,x) = -(15*n^2-90*n+27-(162*n^2-276*n+162)*x-(288*n^2-576*n+288)*x^2)*P(n-1,x) - (n^2-42*n+5+(9*n^2-60*n-33)*x+(198*n-246)*x^2+(288*n-288)*x^3)*d/dx P(n-1,x) + ((7*n+11)*x+(9*n+9)*x^2)*d^2/dx^2 P(n-1,x) + (72*n^2+36*n-216+(72*n^2+276*n-576)*x-(216*n^2-612*n+432)*x^2)*P(n-2,x) - (16*n-24-144*x-264*x^2+(108*n-144)*x^3)*d/dx P(n-2,x) + (2*n-4-4*x)*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(15*n^2 - 90*n + 27 - (162*n^2 - 276*n + 162)*x - (288*n^2 - 576*n + 288)*x^2)*P[n - 1, x] - (n^2 - 42*n + 5 + (9*n^2 - 60*n - 33)*x + (198*n - 246)*x^2 + (288*n - 288)*x^3)*D[P[n - 1, x], x] + ((7*n + 11)*x + (9*n + 9)*x^2)*D[P[n - 1, x], {x, 2}] + (72*n^2 + 36*n - 216 + (72*n^2 + 276*n - 576)*x - (216*n^2 - 612*n + 432)*x^2)*P[n - 2, x] - (16*n - 24 - 144*x - 264*x^2 + (108*n - 144)*x^3)*D[P[n - 2, x], x] + (2*n - 4 - 4*x)*D[P[n - 2, x], {x, 2}])/(16*n^2 + 8*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(15*n^2 - 90*n + 27 - (162*n^2 - 276*n + 162)*x - (288*n^2 - 576*n + 288)*x^2)*v[i - 1] - (n^2 - 42*n + 5 + (9*n^2 - 60*n - 33)*x + (198*n - 246)*x^2 + (288*n - 288)*x^3)*deriv(v[i - 1], x) + ((7*n + 11)*x + (9*n + 9)*x^2)*deriv(deriv(v[i - 1], x), x) + (72*n^2 + 36*n - 216 + (72*n^2 + 276*n - 576)*x - (216*n^2 - 612*n + 432)*x^2)*v[i - 2] - (16*n - 24 - 144*x - 264*x^2 + (108*n - 144)*x^3)*deriv(v[i - 2], x) + (2*n - 4 - 4*x)*deriv(deriv(v[i - 2], x), x))/(16*n^2 + 8*n)); 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
A330798 triangle, offset 0, from P0
%e ExampleTriangle begins: 0: 1; 1: 2, 2; 2: 6, 15, 9; 3: 20, 84, 112, 48; 4: 70, 420, 900, 825, 275; 5: 252, 1980, 5940, 8580, 6006, 1638; 6: 924, 9009, 35035, 70070, 76440, 43316, 9996; 7: 3432, 40040, 192192, 495040, 742560, 651168, 310080, 62016; 8: 12870, 175032, 1002456, 3174444, 6104700, 7325640, 5372136, 2206413, 389367; 9: 48620, 755820, 5038800, 18992400, 44767800, 68643960, 68643960, 43263000, 15622750, 2466750;
%F FormulaThe row polynomials satisfy (16*n^2+8*n)*P(n,x) = -(15*n^2-90*n+27-(162*n^2-276*n+162)*x-(288*n^2-576*n+288)*x^2)*P(n-1,x) - (n^2-42*n+5+(9*n^2-60*n-33)*x+(198*n-246)*x^2+(288*n-288)*x^3)*d/dx P(n-1,x) + ((7*n+11)*x+(9*n+9)*x^2)*d^2/dx^2 P(n-1,x) + (72*n^2+36*n-216+(72*n^2+276*n-576)*x-(216*n^2-612*n+432)*x^2)*P(n-2,x) - (16*n-24-144*x-264*x^2+(108*n-144)*x^3)*d/dx P(n-2,x) + (2*n-4-4*x)*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(15*n^2 - 90*n + 27 - (162*n^2 - 276*n + 162)*x - (288*n^2 - 576*n + 288)*x^2)*P[n - 1, x] - (n^2 - 42*n + 5 + (9*n^2 - 60*n - 33)*x + (198*n - 246)*x^2 + (288*n - 288)*x^3)*D[P[n - 1, x], x] + ((7*n + 11)*x + (9*n + 9)*x^2)*D[P[n - 1, x], {x, 2}] + (72*n^2 + 36*n - 216 + (72*n^2 + 276*n - 576)*x - (216*n^2 - 612*n + 432)*x^2)*P[n - 2, x] - (16*n - 24 - 144*x - 264*x^2 + (108*n - 144)*x^3)*D[P[n - 2, x], x] + (2*n - 4 - 4*x)*D[P[n - 2, x], {x, 2}])/(16*n^2 + 8*n)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(15*n^2 - 90*n + 27 - (162*n^2 - 276*n + 162)*x - (288*n^2 - 576*n + 288)*x^2)*v[i - 1] - (n^2 - 42*n + 5 + (9*n^2 - 60*n - 33)*x + (198*n - 246)*x^2 + (288*n - 288)*x^3)*deriv(v[i - 1], x) + ((7*n + 11)*x + (9*n + 9)*x^2)*deriv(deriv(v[i - 1], x), x) + (72*n^2 + 36*n - 216 + (72*n^2 + 276*n - 576)*x - (216*n^2 - 612*n + 432)*x^2)*v[i - 2] - (16*n - 24 - 144*x - 264*x^2 + (108*n - 144)*x^3)*deriv(v[i - 2], x) + (2*n - 4 - 4*x)*deriv(deriv(v[i - 2], x), x))/(16*n^2 + 8*n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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)
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Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences)
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Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences)
Proof chains ? JSON
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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
Used in 7
- reversal of Triangle read by rows, interpolating between the central binomial c… Image under a real-rootedness preserver (reversal, shift 0)
- one_plus_derivative of Triangle read by rows, interpolating between the central… Image under a real-rootedness preserver (derivative polynomial, shift 0)
- theta_plus_one_squared of Triangle read by rows, interpolating between the cent… Image under a real-rootedness preserver (euler polynomial, shift 0)
- mobius_x_over_x_plus_one of Triangle read by rows, interpolating between the ce… Image under a real-rootedness preserver (mobius, shift 0)
- theta_plus_one of Triangle read by rows, interpolating between the central bino… Image under a real-rootedness preserver (euler polynomial, shift 0)
- shift_x_plus_one of Triangle read by rows, interpolating between the central bi… Image under a real-rootedness preserver (affine, shift 0)
- derivative of Triangle read by rows, interpolating between the central binomial… Image under a real-rootedness preserver (derivative polynomial, shift 0)
Related ?
Same fingerprint ?
- A104978: Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.
- A108426: Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may…
- A330798 → reverse
References
- [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.