{"id":"n-7e1907415114e6d6","revision":0,"first_index":0,"fingerprint":"f6-f25561f72e77d4fd","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","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":true,"rows":[["1"],["7","1"],["48","14","1"],["324","144","21","1"],["2160","1296","288","28","1"],["14256","10800","3240","480","35","1"],["93312","85536","32400","6480","720","42","1"],["606528","653184","299376","75600","11340","1008","49","1"],["3919104","4852224","2612736","798336","151200","18144","1344","56","1"],["25194240","35271936","21835008","7838208","1796256","272160","27216","1728","63","1"],["161243136","251942400","176359680","72783360","19595520","3592512","453600","38880","2160","70","1"],["1027924992","1773674496","1385683200","646652160","200154240","43110144","6586272","712800","53460","2640","77","1"],["6530347008","12335099904","10642046976","5542732800","1939956480","480370176","86220288","11290752","1069200","71280","3168","84","1"],["41358864384","84894511104","80178149376","46115536896","18013881600","5043886848","1040802048","160123392","18347472","1544400","92664","3744","91","1"],["261213880320","579024101376","594261577728","374164697088","161404379136","50438868480","11769069312","2081604096","280215936","28540512","2162160","117936","4368","98","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","63","490","3773","28812","218491","1647086","12353145","92236816","686011319","5084554482","37569208117","276825744020","2034669218547"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["5","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["36","12","1"],["6","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182592,"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-b8ee2445d4524837"]},{"id":182593,"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-b8ee2445d4524837"]},{"id":182594,"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-b8ee2445d4524837"]},{"id":182595,"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-b8ee2445d4524837"]}],"region_exact":{},"incoming_edges":[{"id":"e-b8ee2445d4524837","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-7e1907415114e6d6","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":"1efff36a0581123f28f0188ef56da9fc8112b2547776f20d6ede9c57c7285e58","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-22693abd431a71f2","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-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-5142caf017d50f0b","kind":"fingerprint_related","node_a_id":"n-7e1907415114e6d6","node_b_id":"n-d8a3f05a6316560e","fingerprint":"f6-f25561f72e77d4fd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f25561f72e77d4fd"},"other_id":"n-d8a3f05a6316560e","other":{"id":"n-d8a3f05a6316560e","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-f25561f72e77d4fd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["192","56","4"],["2592","1152","168","8"],["34560","20736","4608","448","16"]],"card_rows":[["192","56","4"],["2592","1152","168","8"],["34560","20736","4608","448","16"],["456192","345600","103680","15360","1120","32"]],"row_sums_preview":["252","3920","60368","921984","13983424","210827008","3162405120","47225249792"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["D+1"],"edges":["e-1cdbad7f78d15f92"],"text":"A038328 \u2192 D+1"},"headline":"A038328 \u2192 D+1","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:        1;\n  1:        7,        1;\n  2:       48,       14,        1;\n  3:      324,      144,       21,       1;\n  4:     2160,     1296,      288,      28,       1;\n  5:    14256,    10800,     3240,     480,      35,      1;\n  6:    93312,    85536,    32400,    6480,     720,     42,     1;\n  7:   606528,   653184,   299376,   75600,   11340,   1008,    49,    1;\n  8:  3919104,  4852224,  2612736,  798336,  151200,  18144,  1344,   56,  1;\n  9: 25194240, 35271936, 21835008, 7838208, 1796256, 272160, 27216, 1728, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+5+x)*P(n,x) = (6*n+36+(n+12)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 36 + (n + 12)*x + x^2)*P[n - 1, x])/(n + 5 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 36 + (n + 12)*x + x^2)*v[i - 1])/(n + 5 + 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-22693abd431a71f2","steps":["D+1"],"edges":["e-b8ee2445d4524837"],"text":"A038255 \u2192 D+1"},"headline":"A038255 \u2192 D+1","chain":null}