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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":"fa7332782322e3d585984c34257fc5bf6e19288f467c320e8fba78ca2ff18c73","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-563143c7a3f59683","role":"sequence","name":"shift_x_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:      2,       3;\n  2:      6,      18,       13;\n  3:     22,      99,      145,       69;\n  4:     92,     552,     1220,     1176,       417;\n  5:    422,    3165,     9360,    13635,      9780,      2763;\n  6:   2074,   18666,    69165,   135000,    146345,     83526,     19609;\n  7:  10754,  112917,   502929,  1231335,   1789291,   1542933,    731003,    146793;\n  8:  58202,  698424,  3633812, 10703952,  19520606,  22565424,  16145612,   6537264,  1146833;\n  9: 326240, 4404240, 26214060, 90269550, 198160410, 287539560, 275767254, 168552738, 59576718, 9278595;\n","recurrence_sentence":"The row polynomials satisfy (n^2+7*n+12+(3*n^2+21*n+36)*x+(3*n^2+21*n+36)*x^2+(n^2+7*n+12)*x^3)*P(n,x) = (35*n^2-55*n+60+(243*n^2-459*n+396)*x+(693*n^2-1509*n+1116)*x^2+(1043*n^2-2539*n+1716)*x^3+(876*n^2-2328*n+1512)*x^4+(390*n^2-1110*n+720)*x^5+(72*n^2-216*n+144)*x^6)*P(n-1,x) - (36*n-72+(327*n-624)*x+(1242*n-2304)*x^2+(2562*n-4704)*x^3+(3108*n-5736)*x^4+(2223*n-4176)*x^5+(870*n-1680)*x^6+(144*n-288)*x^7)*d/dx P(n-1,x) + (12+126*x+570*x^2+1452*x^3+2280*x^4+2262*x^5+1386*x^6+480*x^7+72*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[((35*n^2 - 55*n + 60 + (243*n^2 - 459*n + 396)*x + (693*n^2 - 1509*n + 1116)*x^2 + (1043*n^2 - 2539*n + 1716)*x^3 + (876*n^2 - 2328*n + 1512)*x^4 + (390*n^2 - 1110*n + 720)*x^5 + (72*n^2 - 216*n + 144)*x^6)*P[n - 1, x] - (36*n - 72 + (327*n - 624)*x + (1242*n - 2304)*x^2 + (2562*n - 4704)*x^3 + (3108*n - 5736)*x^4 + (2223*n - 4176)*x^5 + (870*n - 1680)*x^6 + (144*n - 288)*x^7)*D[P[n - 1, x], x] + (12 + 126*x + 570*x^2 + 1452*x^3 + 2280*x^4 + 2262*x^5 + 1386*x^6 + 480*x^7 + 72*x^8)*D[P[n - 1, x], {x, 2}])/(n^2 + 7*n + 12 + (3*n^2 + 21*n + 36)*x + (3*n^2 + 21*n + 36)*x^2 + (n^2 + 7*n + 12)*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] = ((35*n^2 - 55*n + 60 + (243*n^2 - 459*n + 396)*x + (693*n^2 - 1509*n + 1116)*x^2 + (1043*n^2 - 2539*n + 1716)*x^3 + (876*n^2 - 2328*n + 1512)*x^4 + (390*n^2 - 1110*n + 720)*x^5 + (72*n^2 - 216*n + 144)*x^6)*v[i - 1] - (36*n - 72 + (327*n - 624)*x + (1242*n - 2304)*x^2 + (2562*n - 4704)*x^3 + (3108*n - 5736)*x^4 + (2223*n - 4176)*x^5 + (870*n - 1680)*x^6 + (144*n - 288)*x^7)*deriv(v[i - 1], x) + (12 + 126*x + 570*x^2 + 1452*x^3 + 2280*x^4 + 2262*x^5 + 1386*x^6 + 480*x^7 + 72*x^8)*deriv(deriv(v[i - 1], x), x))/(n^2 + 7*n + 12 + (3*n^2 + 21*n + 36)*x + (3*n^2 + 21*n + 36)*x^2 + (n^2 + 7*n + 12)*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":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-0c759266bd791ca5","e-005169215f5dbacc"],"text":"A056939 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A056939 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}