Real-rooted atlas

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 ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2615930
3208411248264
4704209008252752490
52521980594085806006163824396
69249009350357007076440433169996244790
7343240040192192495040742560651168310080620162496528
81287017503210024563174444610470073256405372136220641338936725763058
94862075582050388001899240044767800686439606864396043263000156227502466750268243860
10184756323323024622290107869080302033424566312670721771050618660900341891550110165055157378652812481870
117054321372879211767536058837680019122246004251887640661404744072108036005408102700266112990077414688010097568029653804824
122704156579462005524204403107364975115155290252968447482054681927300726488462706918937740046126251600204559898405423936700650872404314097641130
1310400600243374040255542742015983947980665997832501955930476504172651683206557024073607600186994406425278860003855167316001556085716643790465207242116280083339741725404
144011660010179587251167658537580438699250372558186000122944201380029857877478005428704996000743496553800076414923585005807534192460316774592316011732392308002643011454002734149780035626286189670

b-files: A330798, A330801 (row sums) ?

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 ?

  • A330798 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A330801

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle 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 Formula
The 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 Mathematica
P[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/GP
P(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 Example
Triangle 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 Formula
The 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 Mathematica
P[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/GP
P(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

Proof chains ? JSON

Used in 7

Same fingerprint ?

References

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