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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"2c0e25330096afee796d70ec8ec20f80ec3ba9118beddbc58aa86fef6108709c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-78ba05b4a3dd925d","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       3,        1;\n   3:      13,        9,        1;\n   4:      63,       73,       18,        1;\n   5:     321,      565,      235,       30,       1;\n   6:    1683,     4261,     2660,      575,      45,       1;\n   7:    8989,    31647,    27811,     8960,    1190,      63,      1;\n   8:   48639,   232849,   276794,   123991,   24430,    2198,     84,    1;\n   9:  265729,  1703205,  2666203,  1590414,  432201,   57498,   3738,  108,   1;\n  10: 1462563, 12412981, 25105500, 19368175, 6937917, 1265817, 121380, 5970, 135, 1;\n","recurrence_sentence":"The row polynomials satisfy -(n-1)*P(n,x) = (n-2+(3*n-6)*x+(n-2)*x^2)*P(n-2,x) + ((3*n-6)*x+(2*n-4)*x^2)*d/dx P(n-2,x) + (n-2)*x^2*d^2/dx^2 P(n-2,x) - (6*n-9+(2*n-3)*x)*P(n-1,x) - (2*n-3)*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (3*n - 6)*x + (n - 2)*x^2)*P[n - 2, x] + ((3*n - 6)*x + (2*n - 4)*x^2)*D[P[n - 2, x], x] + (n - 2)*x^2*D[P[n - 2, x], {x, 2}] - (6*n - 9 + (2*n - 3)*x)*P[n - 1, x] - (2*n - 3)*x*D[P[n - 1, x], x])/(-n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = ((n - 2 + (3*n - 6)*x + (n - 2)*x^2)*v[i - 2] + ((3*n - 6)*x + (2*n - 4)*x^2)*deriv(v[i - 2], x) + (n - 2)*x^2*deriv(deriv(v[i - 2], x), x) - (6*n - 9 + (2*n - 3)*x)*v[i - 1] - (2*n - 3)*x*deriv(v[i - 1], x))/(-n + 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-78ba05b4a3dd925d","steps":["basis A008277"],"edges":["e-907fec0ae54b396c"],"text":"A104684 \u2192 basis A008277"},"headline":"A104684 \u2192 basis A008277","chain":null}