{"id":"n-4ccf1b9afcc98a7b","revision":0,"first_index":0,"fingerprint":"f6-76caa1d365acf455","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","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":true,"rows":[["216","90","9"],["3024","1944","405","27"],["41472","36288","11664","1620","81"],["559872","622080","272160","58320","6075","243"],["7464960","10077696","5598720","1632960","262440","21870","729"],["98537472","156764160","105815808","39191040","8573040","1102248","76545","2187"],["1289945088","2364899328","1881169920","846526464","235146240","41150592","4408992","262440","6561"],["16769286144","34828517376","31926140928","16930529280","5714053632","1269789696","185177664","17006112","885735","19683"],["216710774784","503078584320","522427760640","319261409280","126978969600","34284321792","6348948480","793618560","63772920","2952450","59049"],["2786281390080","7151455567872","8300796641280","5746705367040","2633906626560","838061199360","188563769856","29930757120","3273676560","233834040","9743085","177147"],["35664401793024","100306130042880","128726200221696","99609559695360","51720348303360","18964127711232","5028367196160","969756530688","134688407040","13094706240","841802544","31886460","531441"],["454721122861056","1390911669927936","1955969535836160","1673440602882048","971193207029760","403418716766208","123266830123008","28015188664320","4727563087104","583649763840","51069354336","2984572656","103630995","1594323"],["5777633090469888","19098287160164352","29209145068486656","27383573501706240","17571126330261504","8158022939049984","2823931017363456","739600980738048","147079740487680","22061961073152","2451329008128","194992080192","10446004296","334807830","4782969"],["73183352479285248","259993489071144960","429711461103697920","438137176027299840","308065201894195200","158140136972353536","61185172042874880","18153842254479360","4160255516651520","735398702438400","99278824829184","10028164124160","731220300720","36159245640","1076168025","14348907"],["924421294475182080","3512800919005691904","6239843737707479040","6875383377659166720","5257646112327598080","2957425938184273920","1265121095778828288","419555465436856320","108923053526876160","22188029422141440","3529913771704320","433216690163712","40112656496640","2699890341120","123974556480","3443737680","43046721"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["315","5400","91125","1518750","25059375","410062500","6663515625","107641406250","1729951171875","27679218750000","441137548828125","7006302246093750","110933118896484375","1751575561523437500","27587315093994140625"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["216","90","9"]],"recurrence":{"denominator":{"coeffs":[["5","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["72","30","3"],["12","3","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192223,"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-9f39158fc671eecc"]},{"id":192224,"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-9f39158fc671eecc"]},{"id":192225,"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-9f39158fc671eecc"]},{"id":192226,"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-9f39158fc671eecc"]}],"region_exact":{},"incoming_edges":[{"id":"e-9f39158fc671eecc","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-4ccf1b9afcc98a7b","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":"8b60cc5dd346ba5c92e2e1fde8adb036cd87fc28453a6cad93726b365f5b2b28","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-615df6ea54d3bbef","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(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-1761883a935298b3","kind":"fingerprint_related","node_a_id":"n-956d4c76a0377512","node_b_id":"n-4ccf1b9afcc98a7b","fingerprint":"f6-76caa1d365acf455","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-76caa1d365acf455"},"other_id":"n-956d4c76a0377512","other":{"id":"n-956d4c76a0377512","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-76caa1d365acf455","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","2"],["96","40","4"]],"card_rows":[["1"],["10","2"],["96","40","4"],["896","576","120","8"]],"row_sums_preview":["1","12","140","1600","18000","200000","2200000","24000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013609","D+1"],"edges":["e-13f16c9d95bb60f6","e-8b57720695ad7194"],"text":"A027466 \u2192 basis A013609 \u2192 D+1"},"headline":"A027466 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-73db9281b89ddbed","kind":"fingerprint_related","node_a_id":"n-d02bb449e1edccb4","node_b_id":"n-4ccf1b9afcc98a7b","fingerprint":"f6-76caa1d365acf455","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-76caa1d365acf455"},"other_id":"n-d02bb449e1edccb4","other":{"id":"n-d02bb449e1edccb4","name":"one_plus_derivative of basis_map_A013611 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-76caa1d365acf455","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","4"],["384","160","16"]],"card_rows":[["1"],["20","4"],["384","160","16"],["7168","4608","960","64"]],"row_sums_preview":["1","24","560","12800","288000","6400000","140800000","3072000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013611","D+1"],"edges":["e-9c08db68889738c4","e-1c5385eb34de4de4"],"text":"A027467 \u2192 basis A013611 \u2192 D+1"},"headline":"A027467 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-74ebf2dfcbe28567","kind":"fingerprint_related","node_a_id":"n-4ccf1b9afcc98a7b","node_b_id":"n-d07e742c44ebf2a2","fingerprint":"f6-76caa1d365acf455","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-76caa1d365acf455"},"other_id":"n-d07e742c44ebf2a2","other":{"id":"n-d07e742c44ebf2a2","name":"one_plus_derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-76caa1d365acf455","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["384","160","16"],["7168","4608","960","64"],["131072","114688","36864","5120","256"]],"card_rows":[["384","160","16"],["7168","4608","960","64"],["131072","114688","36864","5120","256"],["2359296","2621440","1146880","245760","25600","1024"]],"row_sums_preview":["560","12800","288000","6400000","140800000","3072000000","66560000000","1433600000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A007318","D+1"],"edges":["e-7b68ca495e98f7ea","e-d550dbaecd32c1b5"],"text":"A038330 \u2192 basis A007318 \u2192 D+1"},"headline":"A038330 \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-d3c5290964446828","kind":"fingerprint_related","node_a_id":"n-0eb9fb88eb406743","node_b_id":"n-4ccf1b9afcc98a7b","fingerprint":"f6-76caa1d365acf455","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-76caa1d365acf455"},"other_id":"n-0eb9fb88eb406743","other":{"id":"n-0eb9fb88eb406743","name":"one_plus_derivative of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-76caa1d365acf455","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["10","2"]],"card_rows":[["1"],["10","2"],["72","30","3"],["448","288","60","4"]],"row_sums_preview":["0","1","12","105","800","5625","37500","240625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["D","D+1"],"edges":["e-62aaaa59f91d42c1","e-70b71aaf69023428"],"text":"A038231 \u2192 D \u2192 D+1"},"headline":"A038231 \u2192 D \u2192 D+1","chain":null}},{"id":"l-e3bdc27e19e59338","kind":"fingerprint_related","node_a_id":"n-98201fe962aff497","node_b_id":"n-4ccf1b9afcc98a7b","fingerprint":"f6-76caa1d365acf455","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-76caa1d365acf455"},"other_id":"n-98201fe962aff497","other":{"id":"n-98201fe962aff497","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-76caa1d365acf455","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","1"],["24","10","1"]],"card_rows":[["1"],["5","1"],["24","10","1"],["112","72","15","1"]],"row_sums_preview":["1","6","35","200","1125","6250","34375","187500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["D+1"],"edges":["e-0ef8ac6724e5260c"],"text":"A038231 \u2192 D+1"},"headline":"A038231 \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:           216,            90,             9;\n  1:          3024,          1944,           405,            27;\n  2:         41472,         36288,         11664,          1620,            81;\n  3:        559872,        622080,        272160,         58320,          6075,          243;\n  4:       7464960,      10077696,       5598720,       1632960,        262440,        21870,          729;\n  5:      98537472,     156764160,     105815808,      39191040,       8573040,      1102248,        76545,        2187;\n  6:    1289945088,    2364899328,    1881169920,     846526464,     235146240,     41150592,      4408992,      262440,       6561;\n  7:   16769286144,   34828517376,   31926140928,   16930529280,    5714053632,   1269789696,    185177664,    17006112,     885735,     19683;\n  8:  216710774784,  503078584320,  522427760640,  319261409280,  126978969600,  34284321792,   6348948480,   793618560,   63772920,   2952450,   59049;\n  9: 2786281390080, 7151455567872, 8300796641280, 5746705367040, 2633906626560, 838061199360, 188563769856, 29930757120, 3273676560, 233834040, 9743085, 177147;\n","recurrence_sentence":"The row polynomials satisfy (n+5+x)*P(n,x) = (12*n+72+(3*n+30)*x+3*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 216 + 90*x + 9*x^2.","mathematica":"P[0, x_] := 216 + 90*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n + 72 + (3*n + 30)*x + 3*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] = 216 + 90*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 72 + (3*n + 30)*x + 3*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-615df6ea54d3bbef","steps":["D+1"],"edges":["e-9f39158fc671eecc"],"text":"A038329 \u2192 D+1"},"headline":"A038329 \u2192 D+1","chain":null}
