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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-c487c167b9898004"]},{"id":185406,"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-c487c167b9898004"]},{"id":185407,"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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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:    0;\n  1:    2,      2;\n  2:    6,     20,      18,        4;\n  3:   14,     94,     204,      188,        70,         6;\n  4:   30,    352,    1350,     2424,      2250,      1056,        210,          8;\n  5:   62,   1194,    7176,    20648,     33060,     30972,      16744,       4776,        558,         10;\n  6:  126,   3852,   34146,   143868,    345660,    512952,     483924,     287736,     102438,      19260,       1386,         12;\n  7:  254,  12086,  152676,   895140,   3002730,   6362466,    8939672,    8483288,    5404914,    2237850,     559812,      72516,      3302,        14;\n  8:  510,  37304,  657630,  5197072,  23307750,  66513624,  128401350,  173132720,  165087450,  110856040,   51277050,   15591216,   2849730,    261128,     7650,     16;\n  9: 1022, 113986, 2766912, 28858160, 168183880, 622876872, 1573979456, 2827216432, 3691297620, 3534020540, 2473396288, 1245753744, 437278088, 101003560, 13834560, 911888, 17374, 18;\n","recurrence_sentence":"The row polynomials satisfy (1+(2*n-3)*x)*P(n,x) = -(2+(4*n+2)*x+(10*n-6)*x^2+(4*n^2+2*n-14)*x^3+(20*n^2-54*n+28)*x^4+(8*n^3-32*n^2+38*n-12)*x^5)*P(n-2,x) - (x+(2*n+1)*x^2+(6*n-6)*x^3+(4*n^2-10*n+2)*x^4-(6*n-5)*x^5-(4*n^2-8*n+3)*x^6)*d/dx P(n-2,x) + (3+(6*n-3)*x+(12*n-15)*x^2+(8*n^2-22*n+15)*x^3)*P(n-1,x) + (x+(2*n-3)*x^2-x^3-(2*n-3)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2 + (4*n + 2)*x + (10*n - 6)*x^2 + (4*n^2 + 2*n - 14)*x^3 + (20*n^2 - 54*n + 28)*x^4 + (8*n^3 - 32*n^2 + 38*n - 12)*x^5)*P[n - 2, x] - (x + (2*n + 1)*x^2 + (6*n - 6)*x^3 + (4*n^2 - 10*n + 2)*x^4 - (6*n - 5)*x^5 - (4*n^2 - 8*n + 3)*x^6)*D[P[n - 2, x], x] + (3 + (6*n - 3)*x + (12*n - 15)*x^2 + (8*n^2 - 22*n + 15)*x^3)*P[n - 1, x] + (x + (2*n - 3)*x^2 - x^3 - (2*n - 3)*x^4)*D[P[n - 1, x], x])/(1 + (2*n - 3)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2 + (4*n + 2)*x + (10*n - 6)*x^2 + (4*n^2 + 2*n - 14)*x^3 + (20*n^2 - 54*n + 28)*x^4 + (8*n^3 - 32*n^2 + 38*n - 12)*x^5)*v[i - 2] - (x + (2*n + 1)*x^2 + (6*n - 6)*x^3 + (4*n^2 - 10*n + 2)*x^4 - (6*n - 5)*x^5 - (4*n^2 - 8*n + 3)*x^6)*deriv(v[i - 2], x) + (3 + (6*n - 3)*x + (12*n - 15)*x^2 + (8*n^2 - 22*n + 15)*x^3)*v[i - 1] + (x + (2*n - 3)*x^2 - x^3 - (2*n - 3)*x^4)*deriv(v[i - 1], x))/(1 + (2*n - 3)*x)); 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":"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}