{"id":"n-da07574fa3209102","revision":0,"first_index":0,"fingerprint":"f6-0205823d5b324a40","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","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"],["36","8"],["486","216","24"],["5832","3888","864","64"],["65610","58320","19440","2880","160"],["708588","787320","349920","77760","8640","384"],["7440174","9920232","5511240","1632960","272160","24192","896"],["76527504","119042784","79361856","29393280","6531840","870912","64512","2048"],["774840978","1377495072","1071385056","476171136","132269760","23514624","2612736","165888","4608"],["7748409780","15496819560","13774950720","7142567040","2380855680","529079040","78382080","7464960","414720","10240"],["76709256822","170465015160","170465015160","101016305280","39284118720","10475764992","1939956480","246343680","20528640","1013760","22528"],["753145430616","1841022163728","2045580181920","1363720121280","606097831680","188563769856","41903059968","6651279360","739031040","54743040","2433024","49152"],["7343167948506","19581781196016","23933288128464","17728361576640","8864180788320","3151708724736","817109669376","155639937024","21616657920","2134978560","142331904","5750784","106496"],["71172243193212","205608702558168","274144936744224","223377355865664","124098531036480","49639412414592","14707974048768","3268438677504","544739779584","67251824640","5977939968","362299392","13418496","229376"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","44","726","10648","146410","1932612","24801854","311794736","3858459858","47158953820","570623341222","6847480094664","81599137794746","966635940030068"],"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":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["9","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186447,"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-0e1fda911995b99e"]},{"id":186448,"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-0e1fda911995b99e"]},{"id":186449,"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-0e1fda911995b99e"]},{"id":186450,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-0e1fda911995b99e"]}],"region_exact":{},"incoming_edges":[{"id":"e-0e1fda911995b99e","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-da07574fa3209102","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":"41458a5d42dddef581155928aff99b7021dae69e277b01ba98cb1e5c3df32f10","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-20a6b63b1c7531be","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","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-5917a9ca9eacbe78","kind":"fingerprint_related","node_a_id":"n-da07574fa3209102","node_b_id":"n-3a5b707c177aa910","fingerprint":"f6-0205823d5b324a40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0205823d5b324a40"},"other_id":"n-3a5b707c177aa910","other":{"id":"n-3a5b707c177aa910","name":"shift_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-0205823d5b324a40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["324","144","16"],["5832","3888","864","64"],["104976","93312","31104","4608","256"]],"card_rows":[["324","144","16"],["5832","3888","864","64"],["104976","93312","31104","4608","256"],["1889568","2099520","933120","207360","23040","1024"]],"row_sums_preview":["484","10648","234256","5153632","113379904","2494357888","54875873536","1207269217792"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-ccf7b0982517b62a","e-0ca963c00c6b2bae"],"text":"A038328 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A038328 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-6e69f6297393b187","kind":"fingerprint_related","node_a_id":"n-da07574fa3209102","node_b_id":"n-24d208fb12b74cc6","fingerprint":"f6-0205823d5b324a40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0205823d5b324a40"},"other_id":"n-24d208fb12b74cc6","other":{"id":"n-24d208fb12b74cc6","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-0205823d5b324a40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","9"],["4","36","81"]],"card_rows":[["1"],["2","9"],["4","36","81"],["8","108","486","729"]],"row_sums_preview":["1","11","121","1331","14641","161051","1771561","19487171"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-366b2525683554ec","steps":["basis A013609","reverse"],"edges":["e-0da8e8e5bf86fd8d","e-fa090f76810df478"],"text":"A038279 \u2192 basis A013609 \u2192 reverse"},"headline":"A038279 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-8a8f690cc221f953","kind":"fingerprint_related","node_a_id":"n-da07574fa3209102","node_b_id":"n-4675eb44920357da","fingerprint":"f6-0205823d5b324a40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0205823d5b324a40"},"other_id":"n-4675eb44920357da","other":{"id":"n-4675eb44920357da","name":"basis_map_A013609 of basis_map_A013609 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-0205823d5b324a40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","4"],["324","144","16"]],"card_rows":[["1"],["18","4"],["324","144","16"],["5832","3888","864","64"]],"row_sums_preview":["1","22","484","10648","234256","5153632","113379904","2494357888"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013609","basis A013609"],"edges":["e-f16669f3777d4fa6","e-cc970da74fc6993c"],"text":"A027467 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A027467 \u2192 basis A013609 \u2192 basis A013609","chain":null}},{"id":"l-f033b31db016ffea","kind":"fingerprint_related","node_a_id":"n-20a6b63b1c7531be","node_b_id":"n-da07574fa3209102","fingerprint":"f6-0205823d5b324a40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0205823d5b324a40"},"other_id":"n-20a6b63b1c7531be","other":{"id":"n-20a6b63b1c7531be","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-0205823d5b324a40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","2"],["81","36","4"]],"card_rows":[["1"],["9","2"],["81","36","4"],["729","486","108","8"]],"row_sums_preview":["1","11","121","1331","14641","161051","1771561","19487171"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-366b2525683554ec","steps":["basis A013609"],"edges":["e-0da8e8e5bf86fd8d"],"text":"A038279 \u2192 basis A013609"},"headline":"A038279 \u2192 basis A013609","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:        36,          8;\n  3:       486,        216,         24;\n  4:      5832,       3888,        864,        64;\n  5:     65610,      58320,      19440,      2880,       160;\n  6:    708588,     787320,     349920,     77760,      8640,      384;\n  7:   7440174,    9920232,    5511240,   1632960,    272160,    24192,     896;\n  8:  76527504,  119042784,   79361856,  29393280,   6531840,   870912,   64512,   2048;\n  9: 774840978, 1377495072, 1071385056, 476171136, 132269760, 23514624, 2612736, 165888, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (9*n+2*n*x)*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[((9*n + 2*n*x)*P[n - 1, x])/(n - 1)]];\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] = ((9*n + 2*n*x)*v[i - 1])/(n - 1)); 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-366b2525683554ec","steps":["basis A013609","D"],"edges":["e-0da8e8e5bf86fd8d","e-0e1fda911995b99e"],"text":"A038279 \u2192 basis A013609 \u2192 D"},"headline":"A038279 \u2192 basis A013609 \u2192 D","chain":null}