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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":"9ab18cfba4d883ec9edbf421bc0ceeb4c72b476234f7580149050937f422250a","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-44fd873021acce78","role":"sequence","name":"basis_map_A013611 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:     3,      7;\n  2:    10,     48,      54;\n  3:    33,    239,     555,      413;\n  4:   109,   1056,    3730,     5688,      3161;\n  5:   360,   4368,   20736,    48112,     54552,     24192;\n  6:  1189,  17334,  103395,   322836,    556379,    501846,    185149;\n  7:  3927,  66853,  480267,  1886641,   4375893,   5992255,   4486281,   1417003;\n  8: 12970, 252516, 2122116, 10051668,  29344536,  54062316,  61382748,  39275100,  10844766;\n  9: 42837, 938753, 9034980, 50112324, 176492358, 409279518, 624903732, 605800500, 338393421, 82998377;\n","recurrence_sentence":"The row polynomials satisfy (16*n+48*n*x+48*n*x^2+16*n*x^3)*P(n,x) = (113*n^2-299*n+234+(782*n^2-2386*n+1860)*x+(2343*n^2-7653*n+5790)*x^2+(3812*n^2-12668*n+9240)*x^3+(3503*n^2-11461*n+8070)*x^4+(1710*n^2-5394*n+3684)*x^5+(345*n^2-1035*n+690)*x^6)*P(n-1,x) - (78*n-164+(838*n-1852)*x+(3646*n-8180)*x^2+(8454*n-18860)*x^3+(11386*n-24940)*x^4+(8962*n-19124)*x^5+(3834*n-7932)*x^6+(690*n-1380)*x^7)*d/dx P(n-1,x) + (13+196*x+1184*x^2+3780*x^3+7090*x^4+8108*x^5+5576*x^6+2124*x^7+345*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[((113*n^2 - 299*n + 234 + (782*n^2 - 2386*n + 1860)*x + (2343*n^2 - 7653*n + 5790)*x^2 + (3812*n^2 - 12668*n + 9240)*x^3 + (3503*n^2 - 11461*n + 8070)*x^4 + (1710*n^2 - 5394*n + 3684)*x^5 + (345*n^2 - 1035*n + 690)*x^6)*P[n - 1, x] - (78*n - 164 + (838*n - 1852)*x + (3646*n - 8180)*x^2 + (8454*n - 18860)*x^3 + (11386*n - 24940)*x^4 + (8962*n - 19124)*x^5 + (3834*n - 7932)*x^6 + (690*n - 1380)*x^7)*D[P[n - 1, x], x] + (13 + 196*x + 1184*x^2 + 3780*x^3 + 7090*x^4 + 8108*x^5 + 5576*x^6 + 2124*x^7 + 345*x^8)*D[P[n - 1, x], {x, 2}])/(16*n + 48*n*x + 48*n*x^2 + 16*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] = ((113*n^2 - 299*n + 234 + (782*n^2 - 2386*n + 1860)*x + (2343*n^2 - 7653*n + 5790)*x^2 + (3812*n^2 - 12668*n + 9240)*x^3 + (3503*n^2 - 11461*n + 8070)*x^4 + (1710*n^2 - 5394*n + 3684)*x^5 + (345*n^2 - 1035*n + 690)*x^6)*v[i - 1] - (78*n - 164 + (838*n - 1852)*x + (3646*n - 8180)*x^2 + (8454*n - 18860)*x^3 + (11386*n - 24940)*x^4 + (8962*n - 19124)*x^5 + (3834*n - 7932)*x^6 + (690*n - 1380)*x^7)*deriv(v[i - 1], x) + (13 + 196*x + 1184*x^2 + 3780*x^3 + 7090*x^4 + 8108*x^5 + 5576*x^6 + 2124*x^7 + 345*x^8)*deriv(deriv(v[i - 1], x), x))/(16*n + 48*n*x + 48*n*x^2 + 16*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-d5c14b9fec875ecc","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-3c5bbf704ea0dfe8","e-1416a0ca6a15bda9"],"text":"A121574 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A121574 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}