{"id":"n-7c4b5a90bdd6f056","revision":0,"first_index":1,"fingerprint":"f6-49c3a6b95fc852bf","name":"derivative of hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["6","2"],["15","30","3"],["28","140","84","4"],["45","420","630","180","5"],["66","990","2772","1980","330","6"],["91","2002","9009","12012","5005","546","7"],["120","3640","24024","51480","40040","10920","840","8"],["153","6120","55692","175032","218790","111384","21420","1224","9"],["190","9690","116280","503880","923780","755820","271320","38760","1710","10"],["231","14630","223839","1279080","3233230","3879876","2238390","596904","65835","2310","11"],["276","21252","403788","2941884","9806280","16224936","13728792","5883768","1211364","106260","3036","12"],["325","29900","690690","6249100","26558675","57946200","67603900","42493880","14060475","2302300","164450","3900","13"],["378","40950","1130220","12432420","65615550","182530530","280816200","243374040","118107990","31081050","4144140","245700","4914","14"],["435","54810","1781325","23411700","150225075","518959350","1017958725","1163381400","778439025","300450150","64382175","7125300","356265","6090","15"],["496","71920","2718576","42073200","322561200","1354757040","3300049200","4808643120","4242920400","2257928400","709634640","126219600","11780496","503440","7440","16"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["8","48","256","1280","6144","28672","131072","589824","2621440","11534336","50331648","218103808","939524096","4026531840","17179869184"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["6","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["-1"],["2"]]},"terms":[{"coeff":{"coeffs":[["1","-5"],["3","1"],["2","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-4"],["0","4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179124,"position":0,"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-9dc9bc8f119dca51"]},{"id":179125,"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-9dc9bc8f119dca51"]},{"id":179126,"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-9dc9bc8f119dca51"]},{"id":179127,"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-9dc9bc8f119dca51"]}],"region_exact":{},"incoming_edges":[{"id":"e-9dc9bc8f119dca51","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-7c4b5a90bdd6f056","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":"abbba1ef28aa80cdb5c10be8c27ba0e30216b5113e5d94111add6dc46b3ede5c","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-dbc23e13bb290987","role":"sequence","name":"hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).","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-4de4b1fabe1a40e8","kind":"fingerprint_related","node_a_id":"n-c30cb06ad75c5e21","node_b_id":"n-7c4b5a90bdd6f056","fingerprint":"f6-49c3a6b95fc852bf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49c3a6b95fc852bf"},"other_id":"n-c30cb06ad75c5e21","other":{"id":"n-c30cb06ad75c5e21","name":"Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","first_index":0,"fingerprint":"f6-49c3a6b95fc852bf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["5","10","1"]],"card_rows":[["1"],["3","1"],["5","10","1"],["7","35","21","1"]],"row_sums_preview":["1","4","16","64","256","1024","4096","16384"],"regions":["(-inf, +inf)","(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A103327"],"derivation":{"root":"n-c30cb06ad75c5e21","steps":[],"edges":[],"text":"A103327"},"headline":"A103327: Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","chain":null}},{"id":"l-525f9f897ecb807a","kind":"fingerprint_related","node_a_id":"n-6ac01ffa07d08c80","node_b_id":"n-7c4b5a90bdd6f056","fingerprint":"f6-49c3a6b95fc852bf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49c3a6b95fc852bf"},"other_id":"n-6ac01ffa07d08c80","other":{"id":"n-6ac01ffa07d08c80","name":"derivative of Triangle read by rows: T(n, k) = binomial(2n, 2k).","first_index":1,"fingerprint":"f6-49c3a6b95fc852bf","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"],["15","30","3"],["28","140","84","4"]],"row_sums_preview":["0","1","8","48","256","1280","6144","28672"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["D"],"edges":["e-13f8569d458f0f6e"],"text":"A086645 \u2192 D"},"headline":"A086645 \u2192 D","chain":null}},{"id":"l-650652160c0c82b4","kind":"fingerprint_related","node_a_id":"n-cf148c584bf44042","node_b_id":"n-7c4b5a90bdd6f056","fingerprint":"f6-49c3a6b95fc852bf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49c3a6b95fc852bf"},"other_id":"n-cf148c584bf44042","other":{"id":"n-cf148c584bf44042","name":"derivative of veronese_section_2_0 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-49c3a6b95fc852bf","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"],["15","30","3"],["28","140","84","4"]],"row_sums_preview":["0","1","8","48","256","1280","6144","28672"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["Veronese 2","D"],"edges":["e-f72875fb5d524bf0","e-2f59879ce272350a"],"text":"A034870 \u2192 Veronese 2 \u2192 D"},"headline":"A034870 \u2192 Veronese 2 \u2192 D","chain":null}},{"id":"l-8de4e77e3caf8241","kind":"fingerprint_related","node_a_id":"n-fe90ba2487d0ea92","node_b_id":"n-7c4b5a90bdd6f056","fingerprint":"f6-49c3a6b95fc852bf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49c3a6b95fc852bf"},"other_id":"n-fe90ba2487d0ea92","other":{"id":"n-fe90ba2487d0ea92","name":"reversal of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","first_index":0,"fingerprint":"f6-49c3a6b95fc852bf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","10","5"]],"card_rows":[["1"],["1","3"],["1","10","5"],["1","21","35","7"]],"row_sums_preview":["1","4","16","64","256","1024","4096","16384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["reverse"],"edges":["e-77dbb4f6441b1163"],"text":"A103327 \u2192 reverse"},"headline":"A103327 \u2192 reverse","chain":null}},{"id":"l-c234ace73deff6f6","kind":"fingerprint_related","node_a_id":"n-ff2725515a729738","node_b_id":"n-7c4b5a90bdd6f056","fingerprint":"f6-49c3a6b95fc852bf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49c3a6b95fc852bf"},"other_id":"n-ff2725515a729738","other":{"id":"n-ff2725515a729738","name":"reversal of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","first_index":1,"fingerprint":"f6-49c3a6b95fc852bf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["5","10","1"]],"card_rows":[["1"],["3","1"],["5","10","1"],["7","35","21","1"]],"row_sums_preview":["1","4","16","64","256","1024","4096","16384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba8de4e02acfb6ca","steps":["reverse"],"edges":["e-8c740c1640063617"],"text":"A091042 \u2192 reverse"},"headline":"A091042 \u2192 reverse","chain":null}},{"id":"l-efb245cdce0bfd4b","kind":"fingerprint_related","node_a_id":"n-ba8de4e02acfb6ca","node_b_id":"n-7c4b5a90bdd6f056","fingerprint":"f6-49c3a6b95fc852bf","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49c3a6b95fc852bf"},"other_id":"n-ba8de4e02acfb6ca","other":{"id":"n-ba8de4e02acfb6ca","name":"Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","first_index":1,"fingerprint":"f6-49c3a6b95fc852bf","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","10","5"]],"card_rows":[["1"],["1","3"],["1","10","5"],["1","21","35","7"]],"row_sums_preview":["1","4","16","64","256","1024","4096","16384"],"regions":["(-inf, +inf)","(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A091042"],"derivation":{"root":"n-ba8de4e02acfb6ca","steps":[],"edges":[],"text":"A091042"},"headline":"A091042: Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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:   6,     2;\n   2:  15,    30,      3;\n   3:  28,   140,     84,       4;\n   4:  45,   420,    630,     180,       5;\n   5:  66,   990,   2772,    1980,     330,       6;\n   6:  91,  2002,   9009,   12012,    5005,     546,       7;\n   7: 120,  3640,  24024,   51480,   40040,   10920,     840,      8;\n   8: 153,  6120,  55692,  175032,  218790,  111384,   21420,   1224,     9;\n   9: 190,  9690, 116280,  503880,  923780,  755820,  271320,  38760,  1710,   10;\n  10: 231, 14630, 223839, 1279080, 3233230, 3879876, 2238390, 596904, 65835, 2310, 11;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-n)*P(n,x) = (2*n^2+3*n+1+(6*n^2+n-5)*x)*P(n-1,x) + ((4*n+4)*x-(4*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 6 + 2*x.","mathematica":"P[1, x_] := 6 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 3*n + 1 + (6*n^2 + n - 5)*x)*P[n - 1, x] + ((4*n + 4)*x - (4*n + 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - n)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 6 + 2*x; for(n = 2, N, my(i = n); v[i] = ((2*n^2 + 3*n + 1 + (6*n^2 + n - 5)*x)*v[i - 1] + ((4*n + 4)*x - (4*n + 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - n)); 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-dbc23e13bb290987","steps":["D"],"edges":["e-9dc9bc8f119dca51"],"text":"Hadamard-factor parent of A155495 \u2192 D"},"headline":"Hadamard-factor parent of A155495 \u2192 D","chain":null}
