{"id":"n-957a5bf3e47ae72c","revision":0,"first_index":0,"fingerprint":"f6-a8fa044417626521","name":"derivative of shift_x_plus_one of Triangle of B-analogs of Stirling numbers of 2nd kind.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["6","2"],["38","32","3"],["256","368","132","4"],["1844","3792","2472","504","5"],["14296","38472","36792","14424","1850","6"],["119160","398176","506412","299744","78990","6600","7"],["1063168","4257920","6797088","5445024","2240800","421056","23016","8"],["10107152","47285888","91141824","92654272","52564360","16020192","2225888","78800","9"],["101954912","546506528","1237141440","1529893824","1113794320","476903664","111921264","11773616","265806","10"],["1087209056","6576812288","17127967920","25011837184","22319326640","12405222528","4171130264","773381056","62505126","885840","11"],["12214299648","82379905280","242903742144","409774023872","434197028480","298127932416","131688499936","35715923680","5324369184","333077040","2923052","12"],["144126757696","1073081110272","3536975263104","6777725478528","8331600854000","6831540378240","3758207043584","1356027121600","302384648700","36690220720","1779308520","9566088","13"],["1781336869248","14520059037824","52946549642880","113715206277248","159308383164640","152192334411552","100567576916800","45573956067648","13701334467240","2547977667960","253835309288","9513997128","31089474","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","8","73","760","8617","105840","1409089","20248080","312078385","5130155384","89536277913","1652661724744","32150147751065","657118094918944"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["6","20","15"],["-24","-68","-46"],["28","64","36"],["-8","-16","-8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","14","16","6"],["-12","-40","-44","-16"],["8","24","24","8"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["14","18"],["-20","-24"],["8","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["6","12","6"],["-4","-8","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182517,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 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-7274a75767504b17"]},{"id":182518,"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-7274a75767504b17"]},{"id":182519,"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 real-rootedness)","region_exact":null,"edges":["e-7274a75767504b17"]}],"region_exact":{},"incoming_edges":[{"id":"e-7274a75767504b17","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-957a5bf3e47ae72c","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"391b463055e206947e7e43a4228a386c5d10745f08479e9a0fb8b1b7e4a91b74","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-0c5ba5f813be2bea","role":"sequence","name":"shift_x_plus_one of Triangle of B-analogs of Stirling numbers of 2nd kind.","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:        0;\n  1:        1;\n  2:        6,        2;\n  3:       38,       32,        3;\n  4:      256,      368,      132,        4;\n  5:     1844,     3792,     2472,      504,        5;\n  6:    14296,    38472,    36792,    14424,     1850,        6;\n  7:   119160,   398176,   506412,   299744,    78990,     6600,       7;\n  8:  1063168,  4257920,  6797088,  5445024,  2240800,   421056,   23016,     8;\n  9: 10107152, 47285888, 91141824, 92654272, 52564360, 16020192, 2225888, 78800, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-28*n^2+24*n-6+(16*n^3-64*n^2+68*n-20)*x+(8*n^3-36*n^2+46*n-15)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(24*n^2-40*n+14)*x+(24*n^2-44*n+16)*x^2+(8*n^2-16*n+6)*x^3)*d/dx P(n-2,x) + (8*n^2-20*n+14+(8*n^2-24*n+18)*x)*P(n-1,x) - (4*n-6+(8*n-12)*x+(4*n-6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 28*n^2 + 24*n - 6 + (16*n^3 - 64*n^2 + 68*n - 20)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (24*n^2 - 40*n + 14)*x + (24*n^2 - 44*n + 16)*x^2 + (8*n^2 - 16*n + 6)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 20*n + 14 + (8*n^2 - 24*n + 18)*x)*P[n - 1, x] - (4*n - 6 + (8*n - 12)*x + (4*n - 6)*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 28*n^2 + 24*n - 6 + (16*n^3 - 64*n^2 + 68*n - 20)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (24*n^2 - 40*n + 14)*x + (24*n^2 - 44*n + 16)*x^2 + (8*n^2 - 16*n + 6)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 20*n + 14 + (8*n^2 - 24*n + 18)*x)*v[i - 1] - (4*n - 6 + (8*n - 12)*x + (4*n - 6)*x^2)*deriv(v[i - 1], x))/(2*n - 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-28595a60eb57b37f","steps":["x \u21a6 x+1","D"],"edges":["e-4c7248607accd612","e-7274a75767504b17"],"text":"A039756 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A039756 \u2192 x \u21a6 x+1 \u2192 D","chain":null}