{"id":"n-d07e742c44ebf2a2","revision":0,"first_index":0,"fingerprint":"f6-76caa1d365acf455","name":"one_plus_derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^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":[["384","160","16"],["7168","4608","960","64"],["131072","114688","36864","5120","256"],["2359296","2621440","1146880","245760","25600","1024"],["41943040","56623104","31457280","9175040","1474560","122880","4096"],["738197504","1174405120","792723456","293601280","64225280","8257536","573440","16384"],["12884901888","23622320128","18790481920","8455716864","2348810240","411041792","44040192","2621440","65536"],["223338299392","463856467968","425201762304","225485783040","76101451776","16911433728","2466250752","226492416","11796480","262144"],["3848290697216","8933531975680","9277129359360","5669356830720","2254857830400","608811614208","112742891520","14092861440","1132462080","52428800","1048576"],["65970697666560","169324790677504","196537703464960","136064563937280","62362925137920","19842748907520","4464618504192","708669603840","77510737920","5536481280","230686720","4194304"],["1125899906842624","3166593487994880","4063794976260096","3144603255439360","1632774767247360","598684081324032","158741991260160","30614526885888","4252017623040","413390602240","26575110144","1006632960","16777216"],["19140298416324608","58546795155816448","82331430687866880","70439112921841664","40879842320711680","16980857579372544","5188595371474944","1179226220789760","198994424758272","24567212933120","2149631131648","125627793408","4362076160","67108864"],["324259173170675712","1071856711314178048","1639310264362860544","1536853372840181760","986147580905783296","457854233991970816","158488004074143744","41508762971799552","8254583545528320","1238187531829248","137576392425472","10943576670208","586263035904","18790481920","268435456"],["5476377146882523136","19455550390240542720","32155701339425341440","32786205287257210880","23052800592602726400","11833770970869399552","4578542339919708160","1358468606349803520","311315722288496640","55030556970188800","7429125190975488","750416685957120","54717883351040","2705829396480","80530636800","1073741824"],["92233720368547758080","350488137400481480704","622577612487697367040","685988295241073950720","524579284596115374080","295075847585314897920","126226890355940261888","41860958536408760320","10867748850798428160","2213800691829309440","352195564609208320","43224001111130112","4002222325104640","269380348805120","12369505812480","343597383680","4294967296"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["560","12800","288000","6400000","140800000","3072000000","66560000000","1433600000000","30720000000000","655360000000000","13926400000000000","294912000000000000","6225920000000000000","131072000000000000000","2752512000000000000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["384","160","16"]],"recurrence":{"denominator":{"coeffs":[["5","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["96","40","4"],["16","4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193400,"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-d550dbaecd32c1b5"]},{"id":193401,"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-d550dbaecd32c1b5"]},{"id":193402,"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-d550dbaecd32c1b5"]},{"id":193403,"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-d550dbaecd32c1b5"]}],"region_exact":{},"incoming_edges":[{"id":"e-d550dbaecd32c1b5","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-d07e742c44ebf2a2","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":"568471de796a04fb8477618d314758a76efdf3aee0819052c83d016a1819361d","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-ee879560f84ab2b6","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^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-2ad64536c260f0f0","kind":"fingerprint_related","node_a_id":"n-98201fe962aff497","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-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}},{"id":"l-4304f09a271a7b40","kind":"fingerprint_related","node_a_id":"n-0eb9fb88eb406743","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-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-70f7ddf63f1da051","kind":"fingerprint_related","node_a_id":"n-d02bb449e1edccb4","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-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-4ccf1b9afcc98a7b","other":{"id":"n-4ccf1b9afcc98a7b","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^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":[["216","90","9"],["3024","1944","405","27"],["41472","36288","11664","1620","81"]],"card_rows":[["216","90","9"],["3024","1944","405","27"],["41472","36288","11664","1620","81"],["559872","622080","272160","58320","6075","243"]],"row_sums_preview":["315","5400","91125","1518750","25059375","410062500","6663515625","107641406250"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["D+1"],"edges":["e-9f39158fc671eecc"],"text":"A038329 \u2192 D+1"},"headline":"A038329 \u2192 D+1","chain":null}},{"id":"l-8b9220d8eb99ba83","kind":"fingerprint_related","node_a_id":"n-956d4c76a0377512","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-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}}],"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:            384,             160,              16;\n  1:           7168,            4608,             960,              64;\n  2:         131072,          114688,           36864,            5120,            256;\n  3:        2359296,         2621440,         1146880,          245760,          25600,           1024;\n  4:       41943040,        56623104,        31457280,         9175040,        1474560,         122880,          4096;\n  5:      738197504,      1174405120,       792723456,       293601280,       64225280,        8257536,        573440,        16384;\n  6:    12884901888,     23622320128,     18790481920,      8455716864,     2348810240,      411041792,      44040192,      2621440,       65536;\n  7:   223338299392,    463856467968,    425201762304,    225485783040,    76101451776,    16911433728,    2466250752,    226492416,    11796480,     262144;\n  8:  3848290697216,   8933531975680,   9277129359360,   5669356830720,  2254857830400,   608811614208,  112742891520,  14092861440,  1132462080,   52428800,   1048576;\n  9: 65970697666560, 169324790677504, 196537703464960, 136064563937280, 62362925137920, 19842748907520, 4464618504192, 708669603840, 77510737920, 5536481280, 230686720, 4194304;\n","recurrence_sentence":"The row polynomials satisfy (n+5+x)*P(n,x) = (16*n+96+(4*n+40)*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 384 + 160*x + 16*x^2.","mathematica":"P[0, x_] := 384 + 160*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n + 96 + (4*n + 40)*x + 4*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] = 384 + 160*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 96 + (4*n + 40)*x + 4*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-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}