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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-201482c64bf1b1d4"]},{"id":185607,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-201482c64bf1b1d4"]},{"id":185608,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve 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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"79b5f9e88ad1a9ff78f2d4afa1b4d72765842e4137bef2f7121cf724b1802c00","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6373ae06480cbab5","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       3,       16;\n   3:      13,      128,       144;\n   4:      63,      912,      2160,       1024;\n   5:     321,     6144,     22464,      24576,       6400;\n   6:    1683,    40080,    198720,     348160,     224000,      36864;\n   7:    8989,   256128,   1604880,    3891200,    4128000,    1769472,     200704;\n   8:   48639,  1613584,  12226032,   37954560,   56672000,   41029632,   12644352,    1048576;\n   9:  265729, 10059776,  89429760,  338788352,  654976000,  677117952,  359661568,   83886080,   5308416;\n  10: 1462563, 62218512, 634874112, 2840936448, 6756019200, 9131802624, 7013400576, 2868903936, 525533184, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4+(40*n^2-82*n+42)*x+(144*n^3-464*n^2+500*n-180)*x^2+(192*n^4-800*n^3+1216*n^2-800*n+192)*x^3+(64*n^5-320*n^4+640*n^3-640*n^2+320*n-64)*x^4)*P(n,x) = (28*n-44+(336*n^2-766*n+366)*x+(1200*n^3-2912*n^2+1292*n+420)*x^2+(1472*n^4-2560*n^3-2208*n^2+4448*n-896)*x^3+(448*n^5+1344*n^4-8256*n^3+9536*n^2-2880*n+64)*x^4+(768*n^5-2304*n^4+1280*n^3+1280*n^2-1024*n)*x^5)*P(n-1,x) - (1+(24*n-18)*x+(120*n^2-28*n-96)*x^2+(192*n^3+400*n^2-728*n+48)*x^3+(64*n^4+1088*n^3-1024*n^2-640*n+320)*x^4+(384*n^4+1152*n^3-2944*n^2+1280*n)*x^5+(512*n^4-512*n^3-512*n^2+512*n)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((28*n - 44 + (336*n^2 - 766*n + 366)*x + (1200*n^3 - 2912*n^2 + 1292*n + 420)*x^2 + (1472*n^4 - 2560*n^3 - 2208*n^2 + 4448*n - 896)*x^3 + (448*n^5 + 1344*n^4 - 8256*n^3 + 9536*n^2 - 2880*n + 64)*x^4 + (768*n^5 - 2304*n^4 + 1280*n^3 + 1280*n^2 - 1024*n)*x^5)*P[n - 1, x] - (1 + (24*n - 18)*x + (120*n^2 - 28*n - 96)*x^2 + (192*n^3 + 400*n^2 - 728*n + 48)*x^3 + (64*n^4 + 1088*n^3 - 1024*n^2 - 640*n + 320)*x^4 + (384*n^4 + 1152*n^3 - 2944*n^2 + 1280*n)*x^5 + (512*n^4 - 512*n^3 - 512*n^2 + 512*n)*x^6)*D[P[n - 1, x], x])/(4*n - 4 + (40*n^2 - 82*n + 42)*x + (144*n^3 - 464*n^2 + 500*n - 180)*x^2 + (192*n^4 - 800*n^3 + 1216*n^2 - 800*n + 192)*x^3 + (64*n^5 - 320*n^4 + 640*n^3 - 640*n^2 + 320*n - 64)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((28*n - 44 + (336*n^2 - 766*n + 366)*x + (1200*n^3 - 2912*n^2 + 1292*n + 420)*x^2 + (1472*n^4 - 2560*n^3 - 2208*n^2 + 4448*n - 896)*x^3 + (448*n^5 + 1344*n^4 - 8256*n^3 + 9536*n^2 - 2880*n + 64)*x^4 + (768*n^5 - 2304*n^4 + 1280*n^3 + 1280*n^2 - 1024*n)*x^5)*v[i - 1] - (1 + (24*n - 18)*x + (120*n^2 - 28*n - 96)*x^2 + (192*n^3 + 400*n^2 - 728*n + 48)*x^3 + (64*n^4 + 1088*n^3 - 1024*n^2 - 640*n + 320)*x^4 + (384*n^4 + 1152*n^3 - 2944*n^2 + 1280*n)*x^5 + (512*n^4 - 512*n^3 - 512*n^2 + 512*n)*x^6)*deriv(v[i - 1], x))/(4*n - 4 + (40*n^2 - 82*n + 42)*x + (144*n^3 - 464*n^2 + 500*n - 180)*x^2 + (192*n^4 - 800*n^3 + 1216*n^2 - 800*n + 192)*x^3 + (64*n^5 - 320*n^4 + 640*n^3 - 640*n^2 + 320*n - 64)*x^4)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-78ba05b4a3dd925d","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-99f7e1fd97b5ca9c","e-201482c64bf1b1d4"],"text":"A104684 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A104684 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}