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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-b14510ed4e9b5287"]},{"id":185409,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-b14510ed4e9b5287"]},{"id":185410,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve 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n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"879140b163b6bd85c7d03e8a7f950ee323dd31bdbe0fbbd1b3f849a3522e4d5b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b3367e055aead099","role":"sequence","name":"Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.","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,      4,       1;\n  2:    7,     26,      28,       10,         1;\n  3:   15,    108,     251,      256,       117,        20,          1;\n  4:   31,    382,    1526,     2874,      2856,      1506,        386,         38,          1;\n  5:   63,   1256,    7773,    23040,     38222,     37584,      21906,       7168,       1155,         72,          1;\n  6:  127,   3978,   36072,   155250,    381627,    582084,     569416,     356868,     138405,      30642,       3312,        138,         1;\n  7:  255,  12340,  158719,   946032,   3226515,   6963012,   10000083,    9760384,    6465325,    2838396,     783597,     123408,      9345,       268,        1;\n  8:  511,  37814,  676282,  5416282,  24607018,  71175174,  139486954,  191475770,  186729040,  129199090,   62362654,   20252766,   4148998,    480338,    26302,     526,     1;\n  9: 1023, 115008, 2823905, 29780464, 175398420, 656513648, 1677792268, 3052070640, 4044699674, 3944164720, 2826798342, 1470607952, 541090900, 134640336, 21049100, 1834192, 74367, 1040, 1;\n","recurrence_sentence":"The row polynomials satisfy (48*n^2-144*n+72)*P(n,x) = -(192*n^2-540*n+246-(96*n^2-440*n+623)*x-(192*n^3-1000*n^2+1800*n-946)*x^2+(192*n^3-808*n^2+910*n-48)*x^3+(192*n^4-1248*n^3+2912*n^2-2604*n+648)*x^4+(32*n^3-144*n^2+270*n-189)*x^5)*P(n-2,x) + (36*n-18+(48*n^2-156*n+183)*x+(120*n^2-460*n+333)*x^2-(224*n^2-770*n+792)*x^3+(48*n^2-164*n+98)*x^4+(160*n^3-784*n^2+1074*n-147)*x^5+(16*n^2-48*n+63)*x^6)*d/dx P(n-2,x) + ((12*n-6)*x+(48*n^2-144*n+103)*x^2-(12*n+30)*x^3-(16*n^2-48*n+94)*x^4+36*x^5-(32*n^2-96*n+9)*x^6)*d^2/dx^2 P(n-2,x) + (192*n^2-540*n+258+(96*n^2-236*n-37)*x+(192*n^3-784*n^2+912*n-354)*x^2+(16*n^2-48*n+63)*x^3)*P(n-1,x) - (12*n-6+31*x-36*x^2+(80*n^2-240*n+81)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(192*n^2 - 540*n + 246 - (96*n^2 - 440*n + 623)*x - (192*n^3 - 1000*n^2 + 1800*n - 946)*x^2 + (192*n^3 - 808*n^2 + 910*n - 48)*x^3 + (192*n^4 - 1248*n^3 + 2912*n^2 - 2604*n + 648)*x^4 + (32*n^3 - 144*n^2 + 270*n - 189)*x^5)*P[n - 2, x] + (36*n - 18 + (48*n^2 - 156*n + 183)*x + (120*n^2 - 460*n + 333)*x^2 - (224*n^2 - 770*n + 792)*x^3 + (48*n^2 - 164*n + 98)*x^4 + (160*n^3 - 784*n^2 + 1074*n - 147)*x^5 + (16*n^2 - 48*n + 63)*x^6)*D[P[n - 2, x], x] + ((12*n - 6)*x + (48*n^2 - 144*n + 103)*x^2 - (12*n + 30)*x^3 - (16*n^2 - 48*n + 94)*x^4 + 36*x^5 - (32*n^2 - 96*n + 9)*x^6)*D[P[n - 2, x], {x, 2}] + (192*n^2 - 540*n + 258 + (96*n^2 - 236*n - 37)*x + (192*n^3 - 784*n^2 + 912*n - 354)*x^2 + (16*n^2 - 48*n + 63)*x^3)*P[n - 1, x] - (12*n - 6 + 31*x - 36*x^2 + (80*n^2 - 240*n + 81)*x^3)*D[P[n - 1, x], x])/(48*n^2 - 144*n + 72)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(192*n^2 - 540*n + 246 - (96*n^2 - 440*n + 623)*x - (192*n^3 - 1000*n^2 + 1800*n - 946)*x^2 + (192*n^3 - 808*n^2 + 910*n - 48)*x^3 + (192*n^4 - 1248*n^3 + 2912*n^2 - 2604*n + 648)*x^4 + (32*n^3 - 144*n^2 + 270*n - 189)*x^5)*v[i - 2] + (36*n - 18 + (48*n^2 - 156*n + 183)*x + (120*n^2 - 460*n + 333)*x^2 - (224*n^2 - 770*n + 792)*x^3 + (48*n^2 - 164*n + 98)*x^4 + (160*n^3 - 784*n^2 + 1074*n - 147)*x^5 + (16*n^2 - 48*n + 63)*x^6)*deriv(v[i - 2], x) + ((12*n - 6)*x + (48*n^2 - 144*n + 103)*x^2 - (12*n + 30)*x^3 - (16*n^2 - 48*n + 94)*x^4 + 36*x^5 - (32*n^2 - 96*n + 9)*x^6)*deriv(deriv(v[i - 2], x), x) + (192*n^2 - 540*n + 258 + (96*n^2 - 236*n - 37)*x + (192*n^3 - 784*n^2 + 912*n - 354)*x^2 + (16*n^2 - 48*n + 63)*x^3)*v[i - 1] - (12*n - 6 + 31*x - 36*x^2 + (80*n^2 - 240*n + 81)*x^3)*deriv(v[i - 1], x))/(48*n^2 - 144*n + 72)); 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":null,"headline":"one_plus_derivative of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.","chain":null}