{"id":"n-236c1003708dee15","revision":0,"first_index":0,"fingerprint":"f6-8deca67d09eebe9e","name":"derivative of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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":[[],["1"],["30","2"],["675","90","3"],["13500","2700","180","4"],["253125","67500","6750","300","5"],["4556250","1518750","202500","13500","450","6"],["79734375","31893750","5315625","472500","23625","630","7"],["1366875000","637875000","127575000","14175000","945000","37800","840","8"],["23066015625","12301875000","2870437500","382725000","31893750","1701000","56700","1080","9"],["384433593750","230660156250","61509375000","9568125000","956812500","63787500","2835000","81000","1350","10"],["6343154296875","4228769531250","1268630859375","225534375000","26312343750","2104987500","116943750","4455000","111375","1650","11"],["103797070312500","76117851562500","25372617187500","5074523437500","676603125000","63149625000","4209975000","200475000","6682500","148500","1980","12"],["1686702392578125","1349361914062500","494766035156250","109948007812500","16492201171875","1759168125000","136824187500","7818525000","325771875","9652500","193050","2340","13"],["27246730957031250","23613833496093750","9445533398437500","2308908164062500","384818027343750","46178163281250","4104725625000","273648375000","13682418750","506756250","13513500","245700","2730","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","32","768","16384","327680","6291456","117440512","2147483648","38654705664","687194767360","12094627905536","211106232532992","3659174697238528","63050394783186944"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["15","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190108,"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-b8062bd651f856c8"]},{"id":190109,"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-b8062bd651f856c8"]},{"id":190110,"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-b8062bd651f856c8"]},{"id":190111,"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-b8062bd651f856c8"]}],"region_exact":{},"incoming_edges":[{"id":"e-b8062bd651f856c8","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-236c1003708dee15","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":"f296a1cb1483d7d54384f6f194cd61fae7c0a30f8548ebdac060d8e7b8cfa9ca","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-ff0214c333bfdcca","role":"sequence","name":"Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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-2d5b694b5839ec50","kind":"fingerprint_related","node_a_id":"n-236c1003708dee15","node_b_id":"n-ff73176658f063e0","fingerprint":"f6-8deca67d09eebe9e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8deca67d09eebe9e"},"other_id":"n-ff73176658f063e0","other":{"id":"n-ff73176658f063e0","name":"reversal of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-8deca67d09eebe9e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","15"],["1","30","225"]],"card_rows":[["1"],["1","15"],["1","30","225"],["1","45","675","3375"]],"row_sums_preview":["1","16","256","4096","65536","1048576","16777216","268435456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["reverse"],"edges":["e-d8eb2b23fe746b9c"],"text":"A027467 \u2192 reverse"},"headline":"A027467 \u2192 reverse","chain":null}},{"id":"l-5152b3c6daf12171","kind":"fingerprint_related","node_a_id":"n-ff0214c333bfdcca","node_b_id":"n-236c1003708dee15","fingerprint":"f6-8deca67d09eebe9e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8deca67d09eebe9e"},"other_id":"n-ff0214c333bfdcca","other":{"id":"n-ff0214c333bfdcca","name":"Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-8deca67d09eebe9e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","1"],["225","30","1"]],"card_rows":[["1"],["15","1"],["225","30","1"],["3375","675","45","1"]],"row_sums_preview":["1","16","256","4096","65536","1048576","16777216","268435456"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A027467"],"derivation":{"root":"n-ff0214c333bfdcca","steps":[],"edges":[],"text":"A027467"},"headline":"A027467: Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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:           1;\n  2:          30,           2;\n  3:         675,          90,          3;\n  4:       13500,        2700,        180,         4;\n  5:      253125,       67500,       6750,       300,        5;\n  6:     4556250,     1518750,     202500,     13500,      450,       6;\n  7:    79734375,    31893750,    5315625,    472500,    23625,     630,     7;\n  8:  1366875000,   637875000,  127575000,  14175000,   945000,   37800,   840,    8;\n  9: 23066015625, 12301875000, 2870437500, 382725000, 31893750, 1701000, 56700, 1080, 9;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (15*n+n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((15*n + 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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((15*n + 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-ff0214c333bfdcca","steps":["D"],"edges":["e-b8062bd651f856c8"],"text":"A027467 \u2192 D"},"headline":"A027467 \u2192 D","chain":null}
