{"id":"n-d37de8e23475b650","revision":0,"first_index":0,"fingerprint":"f6-2f3702dc82e1aa6a","name":"one_plus_derivative of basis_map_A007318 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["5","2"],["22","18","3"],["95","120","42","4"],["406","700","390","80","5"],["1722","3780","2940","980","135","6"],["7260","19404","19530","9240","2100","210","7"],["30459","96096","119196","73920","24150","4032","308","8"],["127270","463320","684684","528528","228690","55440","7140","432","9"],["529958","2187900","3758040","3483480","1891890","612612","115500","11880","585","10"],["2200276","10161580","19909890","21587280","14174160","5801796","1471470","223080","18810","770","11"],["9111830","46558512","102539580","127481640","98455500","49008960","15807792","3243240","405405","28600","990","12"],["37650172","210924168","516023508","724243520","644336550","379119312","149477328","39207168","6666660","700700","42042","1248","13"],["155266100","946454600","2547314952","3985925944","4018904890","2735312580","1280359080","412375392","90053964","12932920","1161160","60060","1547","14"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","7","43","261","1581","9563","57751","348169","2095513","12591855","75549123","452642061","2708392389","16186123203"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["1","3","-1"],["0","-5","1"],["-2","2","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["-6","-10","-2","2"],["16","30","8","-5"],["-8","-25","-12","3"],["-6","2","6","0"],["4","3","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-3","0","1"],["0","4","8","2","-2"],["0","0","-4","-4","0"],["0","-2","-2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177187,"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-3608920d0997386f"]},{"id":177188,"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-3608920d0997386f"]},{"id":177189,"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-3608920d0997386f"]},{"id":177190,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-3608920d0997386f"]}],"region_exact":{},"incoming_edges":[{"id":"e-3608920d0997386f","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-d37de8e23475b650","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":"8ce918c31df6de0fffd315ae031292d43de8eff95ded8ded9e9b8e7ec57e3a97","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c2b0d68b9e568dde","role":"sequence","name":"basis_map_A007318 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","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:      1;\n  2:      5,      2;\n  3:     22,     18,      3;\n  4:     95,    120,     42,      4;\n  5:    406,    700,    390,     80,      5;\n  6:   1722,   3780,   2940,    980,    135,     6;\n  7:   7260,  19404,  19530,   9240,   2100,   210,    7;\n  8:  30459,  96096, 119196,  73920,  24150,  4032,  308,   8;\n  9: 127270, 463320, 684684, 528528, 228690, 55440, 7140, 432, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^4-2*n^3+n+(2*n^3-5*n^2+3*n)*x+(n^2-n)*x^2)*P(n,x) = (4*n^4-6*n^3-8*n^2+16*n-6+(3*n^4+2*n^3-25*n^2+30*n-10)*x+(6*n^3-12*n^2+8*n-2)*x^2+(3*n^2-5*n+2)*x^3)*P(n-1,x) - ((2*n^3-4*n+2)*x+(2*n^3+4*n^2-8*n+3)*x^2+(4*n^2-2*n)*x^3+(2*n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^4 - 6*n^3 - 8*n^2 + 16*n - 6 + (3*n^4 + 2*n^3 - 25*n^2 + 30*n - 10)*x + (6*n^3 - 12*n^2 + 8*n - 2)*x^2 + (3*n^2 - 5*n + 2)*x^3)*P[n - 1, x] - ((2*n^3 - 4*n + 2)*x + (2*n^3 + 4*n^2 - 8*n + 3)*x^2 + (4*n^2 - 2*n)*x^3 + (2*n - 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 2*n^3 + n + (2*n^3 - 5*n^2 + 3*n)*x + (n^2 - n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^4 - 6*n^3 - 8*n^2 + 16*n - 6 + (3*n^4 + 2*n^3 - 25*n^2 + 30*n - 10)*x + (6*n^3 - 12*n^2 + 8*n - 2)*x^2 + (3*n^2 - 5*n + 2)*x^3)*v[i - 1] - ((2*n^3 - 4*n + 2)*x + (2*n^3 + 4*n^2 - 8*n + 3)*x^2 + (4*n^2 - 2*n)*x^3 + (2*n - 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 2*n^3 + n + (2*n^3 - 5*n^2 + 3*n)*x + (n^2 - n)*x^2)); 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-b1b19325e0094118","steps":["x \u21a6 x+1","D+1"],"edges":["e-0de802f77814f187","e-3608920d0997386f"],"text":"A367270 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A367270 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}