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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":"7344f93bd47018921135fa7738f794836fb05a8c280db587a9593a2d871e6e5c","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-2a476baf987bc613","role":"sequence","name":"basis_map_A013611 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).","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:     4,       8;\n  2:    13,      58,       61;\n  3:    43,     297,      657,      467;\n  4:   142,    1328,     4524,     6656,      3574;\n  5:   469,    5533,    25522,    57530,     63401,     27353;\n  6:  1549,   22062,   128499,   391684,    659043,    580590,    209341;\n  7:  5116,   85376,   600996,  2312872,   5255108,   7050480,   5173276,   1602152;\n  8: 16897,  323296,  2669236, 12418576,  35607070,  64430464,  71861284,  45178384,  12261769;\n  9: 55807, 1204239, 11409612, 62286108, 215888562, 492686370, 740348796, 706458348, 388513287, 93843143;\n","recurrence_sentence":"The row polynomials satisfy (16*n+16+(48*n+48)*x+(48*n+48)*x^2+(16*n+16)*x^3)*P(n,x) = (113*n^2-431*n+446+(782*n^2-3190*n+3048)*x+(2343*n^2-9549*n+8358)*x^2+(3812*n^2-14852*n+11936)*x^3+(3503*n^2-12697*n+9450)*x^4+(1710*n^2-5670*n+3960)*x^5+(345*n^2-1035*n+690)*x^6)*P(n-1,x) - (78*n-216+(838*n-2324)*x+(3646*n-9824)*x^2+(8454*n-21756)*x^3+(11386*n-27704)*x^4+(8962*n-20492)*x^5+(3834*n-8208)*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 - 431*n + 446 + (782*n^2 - 3190*n + 3048)*x + (2343*n^2 - 9549*n + 8358)*x^2 + (3812*n^2 - 14852*n + 11936)*x^3 + (3503*n^2 - 12697*n + 9450)*x^4 + (1710*n^2 - 5670*n + 3960)*x^5 + (345*n^2 - 1035*n + 690)*x^6)*P[n - 1, x] - (78*n - 216 + (838*n - 2324)*x + (3646*n - 9824)*x^2 + (8454*n - 21756)*x^3 + (11386*n - 27704)*x^4 + (8962*n - 20492)*x^5 + (3834*n - 8208)*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 + 16 + (48*n + 48)*x + (48*n + 48)*x^2 + (16*n + 16)*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 - 431*n + 446 + (782*n^2 - 3190*n + 3048)*x + (2343*n^2 - 9549*n + 8358)*x^2 + (3812*n^2 - 14852*n + 11936)*x^3 + (3503*n^2 - 12697*n + 9450)*x^4 + (1710*n^2 - 5670*n + 3960)*x^5 + (345*n^2 - 1035*n + 690)*x^6)*v[i - 1] - (78*n - 216 + (838*n - 2324)*x + (3646*n - 9824)*x^2 + (8454*n - 21756)*x^3 + (11386*n - 27704)*x^4 + (8962*n - 20492)*x^5 + (3834*n - 8208)*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 + 16 + (48*n + 48)*x + (48*n + 48)*x^2 + (16*n + 16)*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-d6b7687ba0ff59dd","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-3771d6044a0ef815","e-e14e4eeffedcfa6e"],"text":"A116412 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A116412 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}