{"id":"n-6474b315bc694992","revision":1,"first_index":0,"fingerprint":"f6-72ae8e6419635e63","name":"derivative of Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.","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":true,"rows":[[],["2"],["4","6"],["6","18","12"],["8","36","48","20"],["10","60","120","100","30"],["12","90","240","300","180","42"],["14","126","420","700","630","294","56"],["16","168","672","1400","1680","1176","448","72"],["18","216","1008","2520","3780","3528","2016","648","90"],["20","270","1440","4200","7560","8820","6720","3240","900","110"],["22","330","1980","6600","13860","19404","18480","11880","4950","1210","132"],["24","396","2640","9900","23760","38808","44352","35640","19800","7260","1584","156"],["26","468","3432","14300","38610","72072","96096","92664","64350","31460","10296","2028","182"],["28","546","4368","20020","60060","126126","192192","216216","180180","110110","48048","14196","2548","210"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","10","36","112","320","864","2240","5632","13824","33280","78848","184320","425984","974848"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-2","0"],["2","-1"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["2","3","1"],["0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161410,"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-c370da7cbe42718e"]},{"id":161411,"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-c370da7cbe42718e"]},{"id":161412,"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-c370da7cbe42718e"]}],"region_exact":{},"incoming_edges":[{"id":"e-c370da7cbe42718e","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-6474b315bc694992","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":"797022b603ccaaf2d53c1bfac01ed5f5a9181390da61018a5e8811776f0feae4","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-801895951aa20f29","role":"sequence","name":"Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.","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":[{"id":"l-895bab94010bb4ac","kind":"fingerprint_related","node_a_id":"n-f889074884265bc4","node_b_id":"n-6474b315bc694992","fingerprint":"f6-72ae8e6419635e63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72ae8e6419635e63"},"other_id":"n-f889074884265bc4","other":{"id":"n-f889074884265bc4","name":"derivative of Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-72ae8e6419635e63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","2"],["0","2","3"],["0","2","6","4"]],"card_rows":[["0","2"],["0","2","3"],["0","2","6","4"],["0","2","9","12","5"]],"row_sums_preview":["2","5","12","28","64","144","320","704"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":["D"],"edges":["e-23d1d4324cad1a70"],"text":"A159854 \u2192 D"},"headline":"A159854 \u2192 D","chain":null}},{"id":"l-97a592f78f53c7b4","kind":"fingerprint_related","node_a_id":"n-787407680a56b31c","node_b_id":"n-6474b315bc694992","fingerprint":"f6-72ae8e6419635e63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72ae8e6419635e63"},"other_id":"n-787407680a56b31c","other":{"id":"n-787407680a56b31c","name":"Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","first_index":0,"fingerprint":"f6-72ae8e6419635e63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["3","9","6"]],"card_rows":[["1"],["2","3"],["3","9","6"],["4","18","24","10"]],"row_sums_preview":["1","5","18","56","160","432","1120","2816"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A124932"],"derivation":{"root":"n-c8c19eba9d68881f","steps":["(1/2+(1/2)n)\u03b8+1+n"],"edges":["e-c94c2672fb190cd9"],"text":"A007318 \u2192 (1/2+(1/2)n)\u03b8+1+n"},"headline":"A124932: Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","chain":"A007318 \u2192 (1/2+(1/2)n)\u03b8+1+n"}},{"id":"l-fb4a9cbc86dbef20","kind":"fingerprint_related","node_a_id":"n-6474b315bc694992","node_b_id":"n-2c11b65ec5b65118","fingerprint":"f6-72ae8e6419635e63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72ae8e6419635e63"},"other_id":"n-2c11b65ec5b65118","other":{"id":"n-2c11b65ec5b65118","name":"reversal of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","first_index":0,"fingerprint":"f6-72ae8e6419635e63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["6","9","3"]],"card_rows":[["1"],["3","2"],["6","9","3"],["10","24","18","4"]],"row_sums_preview":["1","5","18","56","160","432","1120","2816"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["(1/2+(1/2)n)\u03b8+1+n","reverse"],"edges":["e-c94c2672fb190cd9","e-e9ad51cd66b1f493"],"text":"A007318 \u2192 (1/2+(1/2)n)\u03b8+1+n \u2192 reverse"},"headline":"A007318 \u2192 (1/2+(1/2)n)\u03b8+1+n \u2192 reverse","chain":null}}],"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;\n  2:  4,   6;\n  3:  6,  18,   12;\n  4:  8,  36,   48,   20;\n  5: 10,  60,  120,  100,   30;\n  6: 12,  90,  240,  300,  180,   42;\n  7: 14, 126,  420,  700,  630,  294,   56;\n  8: 16, 168,  672, 1400, 1680, 1176,  448,  72;\n  9: 18, 216, 1008, 2520, 3780, 3528, 2016, 648, 90;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2+(n^2-n)*x)*P(n,x) = (2*n+(n^2+3*n)*x+(n^2+n)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + (n^2 + 3*n)*x + (n^2 + n)*x^2)*P[n - 1, x])/(2*n - 2 + (n^2 - n)*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; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + (n^2 + 3*n)*x + (n^2 + n)*x^2)*v[i - 1])/(2*n - 2 + (n^2 - n)*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":{"root":"n-c8c19eba9d68881f","steps":["\u03b8+1","D"],"edges":["e-a112169ce63438eb","e-c370da7cbe42718e"],"text":"A007318 \u2192 \u03b8+1 \u2192 D"},"headline":"A007318 \u2192 \u03b8+1 \u2192 D","chain":null}
