{"id":"n-80e5c51bf45dc008","revision":0,"first_index":0,"fingerprint":"f6-5e3baf07c55bade9","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^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":[[],["11"],["396","242"],["10692","13068","3993"],["256608","470448","287496","58564"],["5773680","14113440","12937320","5270760","805255"],["124711488","381062880","465743520","284621040","86967540","10629366"],["2618941248","9602784576","14670920880","11954083680","5478955020","1339300116","136410197"],["53875362816","230466829824","422522521344","430347012480","262989840960","96429608352","19643068368","1714871048"],["1090976097024","5333660918784","11408108076288","13943243204352","10651088558880","5207198851008","1591088537808","277809109776","21221529219"],["21819521940480","120007370672640","293351350533120","418297296130560","383439188119680","234323948295360","95465312268480","25002819879840","3819875259420","259374246010"],["432026534421504","2640162154798080","7260445925694720","11831837804835840","12653493207949440","9279228352496256","4725532957289760","1650186112069440","378167650682580","51356100709980","3138428376721"],["8483430130458624","57027502543638528","174250702216673280","319459620730567680","390450647559582720","334052220689865216","204143023754917632","89110050051749760","27228070849145760","5546458876677840","677900529371736","37661140520652"],["165426887543943168","1213130508655583232","4077466431870154752","8305950138994759680","11420681441117794560","11166888520204065792","7961577926441787648","4170350342421888768","1592842144675026960","432623792380871520","79314361936493112","8812706881832568","448795257871103"],["3206736589313359872","25475740681767247872","93411049166479908864","209309943502667943936","319779080351298247680","351756988386428072448","286616805351904355328","175154714381719328256","80279244091621358784","27255298919994905760","6662406402665421408","1110401067110903568","113096404983517956","5316497670165374"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","11","638","27753","1073116","38900455","1353735834","45801395717","1517989115192","49524394883139","1595786057345590","50905575229324321","1610467289073169428","50595513998382072863","1580136821795624737106"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["11"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["18","11"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192346,"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-5c805ba05981afbd"]},{"id":192347,"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-5c805ba05981afbd"]},{"id":192348,"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-5c805ba05981afbd"]},{"id":192349,"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-5c805ba05981afbd"]}],"region_exact":{},"incoming_edges":[{"id":"e-5c805ba05981afbd","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-80e5c51bf45dc008","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":"fa6f8a4bd7bea8c038b49a428d3e69ab384300dc012b0295b0f58de77663b1f7","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-77900e9176a9461a","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^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-0711aea69e3bd08b","kind":"fingerprint_related","node_a_id":"n-77900e9176a9461a","node_b_id":"n-80e5c51bf45dc008","fingerprint":"f6-5e3baf07c55bade9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5e3baf07c55bade9"},"other_id":"n-77900e9176a9461a","other":{"id":"n-77900e9176a9461a","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","first_index":0,"fingerprint":"f6-5e3baf07c55bade9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","11"],["324","396","121"]],"card_rows":[["1"],["18","11"],["324","396","121"],["5832","10692","6534","1331"]],"row_sums_preview":["1","29","841","24389","707281","20511149","594823321","17249876309"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b7cf68162d40a956","steps":["x \u21a6 x+1"],"edges":["e-9e4137eebf66e3b4"],"text":"A038277 \u2192 x \u21a6 x+1"},"headline":"A038277 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-169f367ea03ac301","kind":"fingerprint_related","node_a_id":"n-80e5c51bf45dc008","node_b_id":"n-38b5012fcc6605fc","fingerprint":"f6-5e3baf07c55bade9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5e3baf07c55bade9"},"other_id":"n-38b5012fcc6605fc","other":{"id":"n-38b5012fcc6605fc","name":"reversal of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","first_index":0,"fingerprint":"f6-5e3baf07c55bade9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","18"],["121","396","324"]],"card_rows":[["1"],["11","18"],["121","396","324"],["1331","6534","10692","5832"]],"row_sums_preview":["1","29","841","24389","707281","20511149","594823321","17249876309"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["basis A013610","basis A013610"],"edges":["e-32b2129ae50cd22c","e-6ad157539e5db0e9"],"text":"A038220 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038220 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-372d6d6572d598e2","kind":"fingerprint_related","node_a_id":"n-d2377120f9386969","node_b_id":"n-80e5c51bf45dc008","fingerprint":"f6-5e3baf07c55bade9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5e3baf07c55bade9"},"other_id":"n-d2377120f9386969","other":{"id":"n-d2377120f9386969","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-5e3baf07c55bade9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["484","1584","1296"],["10648","52272","85536","46656"],["234256","1533312","3763584","4105728","1679616"]],"card_rows":[["484","1584","1296"],["10648","52272","85536","46656"],["234256","1533312","3763584","4105728","1679616"],["5153632","42166080","137998080","225815040","184757760","60466176"]],"row_sums_preview":["3364","195112","11316496","656356768","38068692544","2207984167552","128063081718016","7427658739644928"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":["basis A013610"],"edges":["e-0be1d4b95988d4e0"],"text":"A038314 \u2192 basis A013610"},"headline":"A038314 \u2192 basis A013610","chain":null}},{"id":"l-96ed169daa00439f","kind":"fingerprint_related","node_a_id":"n-80e5c51bf45dc008","node_b_id":"n-1ab9a5b82290fb1b","fingerprint":"f6-5e3baf07c55bade9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5e3baf07c55bade9"},"other_id":"n-1ab9a5b82290fb1b","other":{"id":"n-1ab9a5b82290fb1b","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-5e3baf07c55bade9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["121","396","324"],["1331","6534","10692","5832"],["14641","95832","235224","256608","104976"]],"card_rows":[["121","396","324"],["1331","6534","10692","5832"],["14641","95832","235224","256608","104976"],["161051","1317690","4312440","7056720","5773680","1889568"]],"row_sums_preview":["841","24389","707281","20511149","594823321","17249876309","500246412961","14507145975869"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["x \u21a6 x/(x+1)"],"edges":["e-301241b1fa993a9c"],"text":"A038321 \u2192 x \u21a6 x/(x+1)"},"headline":"A038321 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-9cb3fdb9fef94670","kind":"fingerprint_related","node_a_id":"n-80e5c51bf45dc008","node_b_id":"n-1f571e6a0df94da8","fingerprint":"f6-5e3baf07c55bade9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5e3baf07c55bade9"},"other_id":"n-1f571e6a0df94da8","other":{"id":"n-1f571e6a0df94da8","name":"basis_map_A013610 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-5e3baf07c55bade9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1936","6336","5184"],["85184","418176","684288","373248"],["3748096","24532992","60217344","65691648","26873856"]],"card_rows":[["1936","6336","5184"],["85184","418176","684288","373248"],["3748096","24532992","60217344","65691648","26873856"],["164916224","1349314560","4415938560","7226081280","5912248320","1934917632"]],"row_sums_preview":["13456","1560896","181063936","21003416576","2436396322816","282621973446656","32784148919812096","3802961274698203136"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A013610","basis A013610"],"edges":["e-889e30cc3b49f76a","e-76c4d64c4a1a3d38"],"text":"A038334 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038334 \u2192 basis A013610 \u2192 basis A013610","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:            11;\n  2:           396,           242;\n  3:         10692,         13068,           3993;\n  4:        256608,        470448,         287496,          58564;\n  5:       5773680,      14113440,       12937320,        5270760,         805255;\n  6:     124711488,     381062880,      465743520,      284621040,       86967540,      10629366;\n  7:    2618941248,    9602784576,    14670920880,    11954083680,     5478955020,    1339300116,     136410197;\n  8:   53875362816,  230466829824,   422522521344,   430347012480,   262989840960,   96429608352,   19643068368,   1714871048;\n  9: 1090976097024, 5333660918784, 11408108076288, 13943243204352, 10651088558880, 5207198851008, 1591088537808, 277809109776, 21221529219;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (18*n+11*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 11.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 11;\nP[n_, x_] := P[n, x] = Expand[Together[((18*n + 11*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] = 11; for(n = 2, N, my(i = n + 1); v[i] = ((18*n + 11*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-b7cf68162d40a956","steps":["x \u21a6 x+1","D"],"edges":["e-9e4137eebf66e3b4","e-5c805ba05981afbd"],"text":"A038277 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A038277 \u2192 x \u21a6 x+1 \u2192 D","chain":null}
