{"id":"n-355f084f1a1b73e3","revision":0,"first_index":0,"fingerprint":"f6-1024c0779b1dc97d","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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":[["60","18"],["900","540","81"],["12000","10800","3240","324"],["150000","180000","81000","16200","1215"],["1800000","2700000","1620000","486000","72900","4374"],["21000000","37800000","28350000","11340000","2551500","306180","15309"],["240000000","504000000","453600000","226800000","68040000","12247200","1224720","52488"],["2700000000","6480000000","6804000000","4082400000","1530900000","367416000","55112400","4723920","177147"],["30000000000","81000000000","97200000000","68040000000","30618000000","9185400000","1837080000","236196000","17714700","590490"],["330000000000","990000000000","1336500000000","1069200000000","561330000000","202078800000","50519700000","8660520000","974308500","64953900","1948617"],["3600000000000","11880000000000","17820000000000","16038000000000","9622800000000","4041576000000","1212472800000","259815600000","38972340000","3897234000","233834040","6377292"],["39000000000000","140400000000000","231660000000000","231660000000000","156370500000000","75057840000000","26270244000000","6755205600000","1266601050000","168880140000","15199212600","829047960","20726199"],["420000000000000","1638000000000000","2948400000000000","3243240000000000","2432430000000000","1313512200000000","525404880000000","157621464000000","35464829400000","5910804900000","709296588000","58033357200","2901667860","66961566"],["4500000000000000","18900000000000000","36855000000000000","44226000000000000","36486450000000000","21891870000000000","9851341500000000","3377602800000000","886620735000000","177324147000000","26598622050000","2901667860000","217625089500","10044234900","215233605"],["48000000000000000","216000000000000000","453600000000000000","589680000000000000","530712000000000000","350269920000000000","175134960000000000","67552056000000000","20265616800000000","4728643920000000","851155905600000","116066714400000","11606671440000","803538792000","34437376800","688747536"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["78","1521","26364","428415","6683274","101362989","1505964408","22024729467","318134981190","4549330231017","64517774185332","908625319776759","12720754476874626","177181937356468005","2456922864676356336"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["60","18"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["20","6"],["10","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191671,"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-04a8061d0e21fde2"]},{"id":191672,"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-04a8061d0e21fde2"]},{"id":191673,"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-04a8061d0e21fde2"]},{"id":191674,"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-04a8061d0e21fde2"]}],"region_exact":{},"incoming_edges":[{"id":"e-04a8061d0e21fde2","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-355f084f1a1b73e3","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":"2f52e32a78f5d0e440a2d2aa481caf04922930c71ae1900ea6f64852c9104cb6","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-4397e742863fdbe2","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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-2a40c97d7c462e73","kind":"fingerprint_related","node_a_id":"n-6a5f903d0d9009ce","node_b_id":"n-355f084f1a1b73e3","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-6a5f903d0d9009ce","other":{"id":"n-6a5f903d0d9009ce","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["60","18"]],"card_rows":[["3"],["60","18"],["900","540","81"],["12000","10800","3240","324"]],"row_sums_preview":["0","3","78","1521","26364","428415","6683274","101362989"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013610","D"],"edges":["e-8d10edd099560dc3","e-ec1a6e1208a933e0"],"text":"A038291 \u2192 basis A013610 \u2192 D"},"headline":"A038291 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-40280dc7ae41a109","kind":"fingerprint_related","node_a_id":"n-355f084f1a1b73e3","node_b_id":"n-80a3aaf82f4475d0","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-80a3aaf82f4475d0","other":{"id":"n-80a3aaf82f4475d0","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","60","100"],["27","270","900","1000"],["81","1080","5400","12000","10000"]],"card_rows":[["9","60","100"],["27","270","900","1000"],["81","1080","5400","12000","10000"],["243","4050","27000","90000","150000","100000"]],"row_sums_preview":["169","2197","28561","371293","4826809","62748517","815730721","10604499373"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["reverse"],"edges":["e-7b98e95757f40f31"],"text":"A038305 \u2192 reverse"},"headline":"A038305 \u2192 reverse","chain":null}},{"id":"l-4a0cfd02aab6f860","kind":"fingerprint_related","node_a_id":"n-6ca7f61b6ad83377","node_b_id":"n-355f084f1a1b73e3","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-6ca7f61b6ad83377","other":{"id":"n-6ca7f61b6ad83377","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","3"],["100","60","9"]],"card_rows":[["1"],["10","3"],["100","60","9"],["1000","900","270","27"]],"row_sums_preview":["1","13","169","2197","28561","371293","4826809","62748517"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013610"],"edges":["e-8d10edd099560dc3"],"text":"A038291 \u2192 basis A013610"},"headline":"A038291 \u2192 basis A013610","chain":null}},{"id":"l-4ea387126a54cb29","kind":"fingerprint_related","node_a_id":"n-fe2b68140d990ced","node_b_id":"n-355f084f1a1b73e3","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-fe2b68140d990ced","other":{"id":"n-fe2b68140d990ced","name":"reversal of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","10"],["9","60","100"]],"card_rows":[["1"],["3","10"],["9","60","100"],["27","270","900","1000"]],"row_sums_preview":["1","13","169","2197","28561","371293","4826809","62748517"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013610","reverse"],"edges":["e-8d10edd099560dc3","e-654768579b54c833"],"text":"A038291 \u2192 basis A013610 \u2192 reverse"},"headline":"A038291 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-66c223130ba54cdf","kind":"fingerprint_related","node_a_id":"n-355f084f1a1b73e3","node_b_id":"n-d1b3a560f252ca5d","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-d1b3a560f252ca5d","other":{"id":"n-d1b3a560f252ca5d","name":"derivative of basis_map_A013611 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["240","800"],["2160","14400","24000"],["17280","172800","576000","640000"]],"card_rows":[["240","800"],["2160","14400","24000"],["17280","172800","576000","640000"],["129600","1728000","8640000","19200000","16000000"]],"row_sums_preview":["1040","40560","1406080","45697600","1425765120","43248208640","1285089628160","37588871623680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013611","D"],"edges":["e-db3287eb465d1cf9","e-d22f3780bc70a92b"],"text":"A013612 \u2192 basis A013611 \u2192 D"},"headline":"A013612 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-9e40bd7d96b6088a","kind":"fingerprint_related","node_a_id":"n-355f084f1a1b73e3","node_b_id":"n-c3a62990f6820c27","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-c3a62990f6820c27","other":{"id":"n-c3a62990f6820c27","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["30","9"],["900","540","81"]],"card_rows":[["1"],["30","9"],["900","540","81"],["27000","24300","7290","729"]],"row_sums_preview":["1","39","1521","59319","2313441","90224199","3518743761","137231006679"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["x \u21a6 x+1","basis A007318"],"edges":["e-62ade03cd9f25f04","e-41449c7eb2a62420"],"text":"A038335 \u2192 x \u21a6 x+1 \u2192 basis A007318"},"headline":"A038335 \u2192 x \u21a6 x+1 \u2192 basis A007318","chain":null}},{"id":"l-b77ef481d67818b8","kind":"fingerprint_related","node_a_id":"n-4397e742863fdbe2","node_b_id":"n-355f084f1a1b73e3","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-4397e742863fdbe2","other":{"id":"n-4397e742863fdbe2","name":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["100","60","9"],["1000","900","270","27"],["10000","12000","5400","1080","81"]],"card_rows":[["100","60","9"],["1000","900","270","27"],["10000","12000","5400","1080","81"],["100000","150000","90000","27000","4050","243"]],"row_sums_preview":["169","2197","28561","371293","4826809","62748517","815730721","10604499373"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":[],"edges":[],"text":"A038305"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","chain":null}},{"id":"l-b9119c8f4480d3ea","kind":"fingerprint_related","node_a_id":"n-a2769223501d377d","node_b_id":"n-355f084f1a1b73e3","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-a2769223501d377d","other":{"id":"n-a2769223501d377d","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["36","240","400"],["216","2160","7200","8000"],["1296","17280","86400","192000","160000"]],"card_rows":[["36","240","400"],["216","2160","7200","8000"],["1296","17280","86400","192000","160000"],["7776","129600","864000","2880000","4800000","3200000"]],"row_sums_preview":["676","17576","456976","11881376","308915776","8031810176","208827064576","5429503678976"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013611"],"edges":["e-db3287eb465d1cf9"],"text":"A013612 \u2192 basis A013611"},"headline":"A013612 \u2192 basis A013611","chain":null}},{"id":"l-fe1d8a1b0eb19831","kind":"fingerprint_related","node_a_id":"n-355f084f1a1b73e3","node_b_id":"n-4e07e76af0d9dd55","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-4e07e76af0d9dd55","other":{"id":"n-4e07e76af0d9dd55","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","240","36"],["8000","7200","2160","216"],["160000","192000","86400","17280","1296"]],"card_rows":[["400","240","36"],["8000","7200","2160","216"],["160000","192000","86400","17280","1296"],["3200000","4800000","2880000","864000","129600","7776"]],"row_sums_preview":["676","17576","456976","11881376","308915776","8031810176","208827064576","5429503678976"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A007318","x \u21a6 x+1"],"edges":["e-37fb8ca4839f71ce","e-061f13a10a85161a"],"text":"A038284 \u2192 basis A007318 \u2192 x \u21a6 x+1"},"headline":"A038284 \u2192 basis A007318 \u2192 x \u21a6 x+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:           60,           18;\n  1:          900,          540,            81;\n  2:        12000,        10800,          3240,           324;\n  3:       150000,       180000,         81000,         16200,         1215;\n  4:      1800000,      2700000,       1620000,        486000,        72900,         4374;\n  5:     21000000,     37800000,      28350000,      11340000,      2551500,       306180,       15309;\n  6:    240000000,    504000000,     453600000,     226800000,     68040000,     12247200,     1224720,      52488;\n  7:   2700000000,   6480000000,    6804000000,    4082400000,   1530900000,    367416000,    55112400,    4723920,    177147;\n  8:  30000000000,  81000000000,   97200000000,   68040000000,  30618000000,   9185400000,  1837080000,  236196000,  17714700,   590490;\n  9: 330000000000, 990000000000, 1336500000000, 1069200000000, 561330000000, 202078800000, 50519700000, 8660520000, 974308500, 64953900, 1948617;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (10*n+20+(3*n+6)*x)*P(n-1,x) for n >= 1, with P(0,x) = 60 + 18*x.","mathematica":"P[0, x_] := 60 + 18*x;\nP[n_, x_] := P[n, x] = Expand[Together[((10*n + 20 + (3*n + 6)*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] = 60 + 18*x; for(n = 1, N, my(i = n + 1); v[i] = ((10*n + 20 + (3*n + 6)*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-4397e742863fdbe2","steps":["D"],"edges":["e-04a8061d0e21fde2"],"text":"A038305 \u2192 D"},"headline":"A038305 \u2192 D","chain":null}
