{"id":"n-47d961ccf2ecbd21","revision":0,"first_index":1,"fingerprint":"f6-cb9ac7fd59880e43","name":"shift_x_plus_one of derivative of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["6","2"],["25","18","3"],["90","102","36","4"],["300","468","258","60","5"],["954","1902","1440","520","90","6"],["2939","7146","6915","3420","915","126","7"],["8850","25414","30024","18960","6930","1470","168","8"],["26195","86796","121278","93492","43575","12600","2212","216","9"],["76500","287364","464004","423220","240660","88620","21168","3168","270","10"],["221016","928296","1701780","1794840","1207430","542052","164682","33480","4365","330","11"],["632916","2939556","6033780","7230340","5625180","2986452","1105272","285480","50490","5830","396","12"],["1799125","9156204","20808894","27941220","24706035","15167520","6635328","2086128","468270","73260","7590","468","13"],["5082270","28127306","70127208","104338512","103419450","72163350","36496152","13556136","3701214","734250","102960","9672","546","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","8","46","232","1091","4912","21468","91824","386373","1604984","6598282","26895704","108850055","437859040"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-4"],["2"]]},"terms":[{"coeff":{"coeffs":[["3","1"],["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","6","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174101,"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-b5f432629a23a91c"]},{"id":174102,"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-6c2ab6aac9125004","e-b5f432629a23a91c"]},{"id":174103,"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-6c2ab6aac9125004","e-b5f432629a23a91c"]},{"id":174104,"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-6c2ab6aac9125004"]},{"id":174105,"position":4,"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-6c2ab6aac9125004"]}],"region_exact":{},"incoming_edges":[{"id":"e-6c2ab6aac9125004","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-47d961ccf2ecbd21","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":"8376281e436ecce05e318b8bd6f3e2edfe20a5b73555e12e7b18d8e8245f713c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e7f9db5cff35db04","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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-b5f432629a23a91c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-47d961ccf2ecbd21","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":"abf68aefd1ab324d2120fdc4d71072f3222f22e04e0f21a954c5bdc823fdd41c","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-1b336857eed4ab2c","role":"sequence","name":"derivative of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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":[{"id":"l-803482d5e9de49a9","kind":"fingerprint_related","node_a_id":"n-ef33eedbc3be092a","node_b_id":"n-47d961ccf2ecbd21","fingerprint":"f6-cb9ac7fd59880e43","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cb9ac7fd59880e43"},"other_id":"n-ef33eedbc3be092a","other":{"id":"n-ef33eedbc3be092a","name":"derivative of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","first_index":0,"fingerprint":"f6-cb9ac7fd59880e43","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["6","2"]],"card_rows":[["1"],["6","2"],["25","18","3"],["90","102","36","4"]],"row_sums_preview":["0","1","8","46","232","1091","4912","21468"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f287bed89e07609f","steps":["D"],"edges":["e-575d1fb5ee5adecf"],"text":"A125662 \u2192 D"},"headline":"A125662 \u2192 D","chain":null}}],"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:     0;\n   2:     1;\n   3:     6,     2;\n   4:    25,    18,      3;\n   5:    90,   102,     36,     4;\n   6:   300,   468,    258,    60,     5;\n   7:   954,  1902,   1440,   520,    90,     6;\n   8:  2939,  7146,   6915,  3420,   915,   126,    7;\n   9:  8850, 25414,  30024, 18960,  6930,  1470,  168,   8;\n  10: 26195, 86796, 121278, 93492, 43575, 12600, 2212, 216, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n-4)*P(n,x) = (3*n+3+(n+1)*x)*P(n-1,x) + (5+6*x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n + 3 + (n + 1)*x)*P[n - 1, x] + (5 + 6*x + x^2)*D[P[n - 1, x], x])/(2*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((3*n + 3 + (n + 1)*x)*v[i - 1] + (5 + 6*x + x^2)*deriv(v[i - 1], x))/(2*n - 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-976cdb6e4a693ec8","steps":["x \u21a6 x+1","D"],"edges":["e-05347468080deadb","e-6c2ab6aac9125004"],"text":"A078812 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A078812 \u2192 x \u21a6 x+1 \u2192 D","chain":null}
