{"id":"n-d57417214bfca832","revision":0,"first_index":1,"fingerprint":"f6-c8df2948d24e794c","name":"shift_x_plus_one of one_plus_derivative of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","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"],["2","1"],["8","9","2"],["44","79","42","6"],["308","765","662","228","24"],["2612","8311","10050","5670","1440","120"],["25988","100989","157382","124620","51960","10440","720"],["296564","1362439","2603202","2664606","1562400","515760","85680","5040"],["3816548","20246445","45857702","57677508","43788024","20346480","5554080","786240","40320"],["54667412","328972471","862110690","1287511350","1202614560","724144680","278359200","64743840","7983360","362880"],["862440068","5805917469","17285512262","29906782140","33203884920","24617499480","12268473840","4018896000","814060800","88905600","3628800"],["14857100084","110645911399","368953375362","725959317006","934543130400","824105457120","505919327760","214915211280","61312204800","10997078400","1077753600","39916800"],["277474957988","2265191981325","8363992922342","18447673161588","27026397074424","27649336239360","20197266471360","10571699033280","3911472512640","988040592000","158988614400","14130547200","479001600"],["5584100659412","49589790516631","200872160746530","490931024632230","806699873659680","939680549447640","796646064614400","496296832711680","226290869925120","74136003701760","16798945363200","2450851603200","199264665600","6227020800"],["120462266974148","1156201277261949","5098019884497542","13676459973927660","24914821165186680","32565344089398120","31445636311449360","22741730339328000","12343654625702400","4983877567468800","1464667940736000","300874077504000","40145603097600","3007651046400","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","19","171","1987","28203","472099","9095691","198113347","4811470443","128876001379","3773325784011","119872143109507","4106182559267883","150854989951869859"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1","2","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","7","9","5","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","1","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","3","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182143,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-6b50b32de0415e08"]},{"id":182144,"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-3f396c8e0bb8dccb","e-6b50b32de0415e08"]},{"id":182145,"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 real-rootedness)","region_exact":null,"edges":["e-3f396c8e0bb8dccb","e-6b50b32de0415e08"]},{"id":182146,"position":3,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 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-3f396c8e0bb8dccb"]}],"region_exact":{},"incoming_edges":[{"id":"e-3f396c8e0bb8dccb","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-d57417214bfca832","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":"b1fb95d56d750301d40aa25335dce8210fd752ba536e04f38c053e828c8a7a75","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-eec36ee0e876d0f2","role":"sequence","name":"shift_x_plus_one of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","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]"}]},{"id":"e-6b50b32de0415e08","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-d57417214bfca832","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"05777c47a9108aeefcabefa2dc21db471867ab678928055f828dc45081f2bf34","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-2eb45ff16c43bd3e","role":"sequence","name":"one_plus_derivative of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","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:        2,         1;\n   3:        8,         9,         2;\n   4:       44,        79,        42,          6;\n   5:      308,       765,       662,        228,         24;\n   6:     2612,      8311,     10050,       5670,       1440,       120;\n   7:    25988,    100989,    157382,     124620,      51960,     10440,       720;\n   8:   296564,   1362439,   2603202,    2664606,    1562400,    515760,     85680,     5040;\n   9:  3816548,  20246445,  45857702,   57677508,   43788024,  20346480,   5554080,   786240,   40320;\n  10: 54667412, 328972471, 862110690, 1287511350, 1202614560, 724144680, 278359200, 64743840, 7983360, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (x+2*x^2+x^3)*P(n-2,x) + (2+7*x+9*x^2+5*x^3+x^4)*d/dx P(n-2,x) + (3+x-x^2)*P(n-1,x) + (2+3*x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(x + 2*x^2 + x^3)*P[n - 2, x] + (2 + 7*x + 9*x^2 + 5*x^3 + x^4)*D[P[n - 2, x], x] + (3 + x - x^2)*P[n - 1, x] + (2 + 3*x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (x + 2*x^2 + x^3)*v[i - 2] + (2 + 7*x + 9*x^2 + 5*x^3 + x^4)*deriv(v[i - 2], x) + (3 + x - x^2)*v[i - 1] + (2 + 3*x + x^2)*deriv(v[i - 1], x)); 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-2cabfd060b04f15d","steps":["x \u21a6 x+1","D+1"],"edges":["e-10a26558e3d19067","e-3f396c8e0bb8dccb"],"text":"A131689 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A131689 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}
