Triangle read by rows: T(n, k) = binomial(n, k)*binomial(2*n+k, k), 0 <= k <= n.

Matches ?

Row sums ? A114496

Submission text ?

Triangle begins:
  0: 1,  10,    15;
  1: 1,  21,    84,     84;
  2: 1,  36,   270,    660,     495;
  3: 1,  55,   660,   2860,    5005,     3003;
  4: 1,  78,  1365,   9100,   27300,    37128,     18564;
  5: 1, 105,  2520,  23800,  107100,   244188,    271320,    116280;
  6: 1, 136,  4284,  54264,  339150,  1139544,   2089164,   1961256,    735471;
  7: 1, 171,  6840, 111720,  921690,  4239774,  11306064,  17305200,  14060475,    4686825;
  8: 1, 210, 10395, 212520, 2231460, 13388760,  48348300, 106563600, 139864725,  100150050,  30045015;
  9: 1, 253, 15180, 379500, 4933500, 37297260, 174053880, 515057400, 965732625, 1108804125, 709634640, 193536720;

The row polynomials satisfy (4*n^2+14*n+12)*P(n,x) = (4*n^2+14*n+12+(18*n^2+57*n+45)*x)*P(n-1,x) + ((8*n+13)*x+(9*n+15)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 10*x + 15*x^2.

P[0, x_] := 1 + 10*x + 15*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 14*n + 12 + (18*n^2 + 57*n + 45)*x)*P[n - 1, x] + ((8*n + 13)*x + (9*n + 15)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 14*n + 12)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 10*x + 15*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 14*n + 12 + (18*n^2 + 57*n + 45)*x)*v[i - 1] + ((8*n + 13)*x + (9*n + 15)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 14*n + 12)); 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)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Iterated real-rootedness preservers
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
  by Row-hypergeometric rows (multiplier sequences)

Proof chains ? JSON

• Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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

Used in 4

• shift_x_plus_one of Number triangle T(n,k) = binomial(n,k)*binomial(2n,n-k). Image under a real-rootedness preserver (reversal, shift 0)
• derivative of Triangle read by rows: T(n, k) = binomial(n, k)*binomial(2*n+k, k… Image under a real-rootedness preserver (derivative polynomial, shift 0)
• shift_x_plus_one of Triangle read by rows: T(n, k) = binomial(n, k)*binomial(2*… Image under a real-rootedness preserver (affine, shift 0)
• mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = binomial(n, k)*bin… Image under a real-rootedness preserver (mobius, shift 0)

Related ?

Same fingerprint ?

• A370258 → reverse

References

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