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for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"98231ddd4242b0d3913c4c6bb5537a00f54a31eaa586ef0673295363efeef7d5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-074c6703c2bee119","role":"sequence","name":"theta_plus_one of Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,   3;\n  2: 1,  10,    12;\n  3: 1,  23,    73,     55;\n  4: 1,  44,   276,    504,     276;\n  5: 1,  75,   815,   2705,    3435,    1477;\n  6: 1, 118,  2045,  10940,   24165,   23442,     8296;\n  7: 1, 175,  4557,  36659,  125111,  205401,   160951,    48393;\n  8: 1, 248,  9268, 106904,  527590, 1302728,  1695484,  1113464,  291010;\n  9: 1, 339, 17526, 279594, 1910496, 6646920, 12771150, 13741170, 7763160, 1794320;\n","recurrence_sentence":"The row polynomials satisfy (n^5+10*n^4+35*n^3+50*n^2+24*n+(3*n^5+30*n^4+105*n^3+150*n^2+72*n)*x+(3*n^5+30*n^4+105*n^3+150*n^2+72*n)*x^2+(n^5+10*n^4+35*n^3+50*n^2+24*n)*x^3)*P(n,x) = (n^5+10*n^4+35*n^3+50*n^2+24*n+(11*n^5+65*n^4+163*n^3+250*n^2+195*n+36)*x+(45*n^5+169*n^4+187*n^3+329*n^2+542*n+168)*x^2+(91*n^5+237*n^4-137*n^3-99*n^2+772*n+336)*x^3+(98*n^5+197*n^4-428*n^3-521*n^2+654*n+360)*x^4+(54*n^5+94*n^4-302*n^3-379*n^2+329*n+204)*x^5+(12*n^5+20*n^4-70*n^3-86*n^2+76*n+48)*x^6)*P(n-1,x) + ((3*n^4+33*n^3+108*n^2+123*n+36)*x-(12*n^4-64*n^3-491*n^2-715*n-228)*x^2-(102*n^4+138*n^3-960*n^2-1878*n-648)*x^3-(228*n^4+532*n^3-1090*n^2-2822*n-1032)*x^4-(237*n^4+623*n^3-800*n^2-2503*n-948)*x^5-(120*n^4+324*n^3-363*n^2-1215*n-468)*x^6-(24*n^4+64*n^3-76*n^2-248*n-96)*x^7)*d/dx P(n-1,x) + ((6*n^3+28*n^2+37*n+12)*x^2+(42*n^3+196*n^2+259*n+84)*x^3+(120*n^3+560*n^2+740*n+240)*x^4+(180*n^3+840*n^2+1110*n+360)*x^5+(150*n^3+700*n^2+925*n+300)*x^6+(66*n^3+308*n^2+407*n+132)*x^7+(12*n^3+56*n^2+74*n+24)*x^8)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (11*n^5 + 65*n^4 + 163*n^3 + 250*n^2 + 195*n + 36)*x + (45*n^5 + 169*n^4 + 187*n^3 + 329*n^2 + 542*n + 168)*x^2 + (91*n^5 + 237*n^4 - 137*n^3 - 99*n^2 + 772*n + 336)*x^3 + (98*n^5 + 197*n^4 - 428*n^3 - 521*n^2 + 654*n + 360)*x^4 + (54*n^5 + 94*n^4 - 302*n^3 - 379*n^2 + 329*n + 204)*x^5 + (12*n^5 + 20*n^4 - 70*n^3 - 86*n^2 + 76*n + 48)*x^6)*P[n - 1, x] + ((3*n^4 + 33*n^3 + 108*n^2 + 123*n + 36)*x - (12*n^4 - 64*n^3 - 491*n^2 - 715*n - 228)*x^2 - (102*n^4 + 138*n^3 - 960*n^2 - 1878*n - 648)*x^3 - (228*n^4 + 532*n^3 - 1090*n^2 - 2822*n - 1032)*x^4 - (237*n^4 + 623*n^3 - 800*n^2 - 2503*n - 948)*x^5 - (120*n^4 + 324*n^3 - 363*n^2 - 1215*n - 468)*x^6 - (24*n^4 + 64*n^3 - 76*n^2 - 248*n - 96)*x^7)*D[P[n - 1, x], x] + ((6*n^3 + 28*n^2 + 37*n + 12)*x^2 + (42*n^3 + 196*n^2 + 259*n + 84)*x^3 + (120*n^3 + 560*n^2 + 740*n + 240)*x^4 + (180*n^3 + 840*n^2 + 1110*n + 360)*x^5 + (150*n^3 + 700*n^2 + 925*n + 300)*x^6 + (66*n^3 + 308*n^2 + 407*n + 132)*x^7 + (12*n^3 + 56*n^2 + 74*n + 24)*x^8)*D[P[n - 1, x], {x, 2}])/(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (3*n^5 + 30*n^4 + 105*n^3 + 150*n^2 + 72*n)*x + (3*n^5 + 30*n^4 + 105*n^3 + 150*n^2 + 72*n)*x^2 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (11*n^5 + 65*n^4 + 163*n^3 + 250*n^2 + 195*n + 36)*x + (45*n^5 + 169*n^4 + 187*n^3 + 329*n^2 + 542*n + 168)*x^2 + (91*n^5 + 237*n^4 - 137*n^3 - 99*n^2 + 772*n + 336)*x^3 + (98*n^5 + 197*n^4 - 428*n^3 - 521*n^2 + 654*n + 360)*x^4 + (54*n^5 + 94*n^4 - 302*n^3 - 379*n^2 + 329*n + 204)*x^5 + (12*n^5 + 20*n^4 - 70*n^3 - 86*n^2 + 76*n + 48)*x^6)*v[i - 1] + ((3*n^4 + 33*n^3 + 108*n^2 + 123*n + 36)*x - (12*n^4 - 64*n^3 - 491*n^2 - 715*n - 228)*x^2 - (102*n^4 + 138*n^3 - 960*n^2 - 1878*n - 648)*x^3 - (228*n^4 + 532*n^3 - 1090*n^2 - 2822*n - 1032)*x^4 - (237*n^4 + 623*n^3 - 800*n^2 - 2503*n - 948)*x^5 - (120*n^4 + 324*n^3 - 363*n^2 - 1215*n - 468)*x^6 - (24*n^4 + 64*n^3 - 76*n^2 - 248*n - 96)*x^7)*deriv(v[i - 1], x) + ((6*n^3 + 28*n^2 + 37*n + 12)*x^2 + (42*n^3 + 196*n^2 + 259*n + 84)*x^3 + (120*n^3 + 560*n^2 + 740*n + 240)*x^4 + (180*n^3 + 840*n^2 + 1110*n + 360)*x^5 + (150*n^3 + 700*n^2 + 925*n + 300)*x^6 + (66*n^3 + 308*n^2 + 407*n + 132)*x^7 + (12*n^3 + 56*n^2 + 74*n + 24)*x^8)*deriv(deriv(v[i - 1], x), x))/(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n + (3*n^5 + 30*n^4 + 105*n^3 + 150*n^2 + 72*n)*x + (3*n^5 + 30*n^4 + 105*n^3 + 150*n^2 + 72*n)*x^2 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-047228a38014e6d4","steps":["\u03b8+1","x \u21a6 x/(x+1)"],"edges":["e-e126ec01ac61762b","e-0b9a69210bfa0e72"],"text":"A056939 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A056939 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)","chain":null}