{"id":"n-7b91a7a037fb3513","revision":3,"first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","name":"Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["5","1"],["25","10","1"],["125","75","15","1"],["625","500","150","20","1"],["3125","3125","1250","250","25","1"],["15625","18750","9375","2500","375","30","1"],["78125","109375","65625","21875","4375","525","35","1"],["390625","625000","437500","175000","43750","7000","700","40","1"],["1953125","3515625","2812500","1312500","393750","78750","10500","900","45","1"],["9765625","19531250","17578125","9375000","3281250","787500","131250","15000","1125","50","1"],["48828125","107421875","107421875","64453125","25781250","7218750","1443750","206250","20625","1375","55","1"],["244140625","585937500","644531250","429687500","193359375","61875000","14437500","2475000","309375","27500","1650","60","1"],["1220703125","3173828125","3808593750","2792968750","1396484375","502734375","134062500","26812500","4021875","446875","35750","1950","65","1"],["6103515625","17089843750","22216796875","17773437500","9775390625","3910156250","1173046875","268125000","46921875","6256250","625625","45500","2275","70","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","36","216","1296","7776","46656","279936","1679616","10077696","60466176","362797056","2176782336","13060694016","78364164096"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A038243","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158465,"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":false,"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-f2f89dae9298aff8"]},{"id":158466,"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-0ab1d5f59b4d29a0","e-868f364735375874","e-8f220a12633250f0","e-f2f89dae9298aff8"]},{"id":158467,"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-0ab1d5f59b4d29a0","e-868f364735375874","e-8f220a12633250f0","e-f2f89dae9298aff8"]},{"id":158468,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-f2f89dae9298aff8"]},{"id":158469,"position":4,"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-868f364735375874","e-8f220a12633250f0"]},{"id":158470,"position":5,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)","region_exact":null,"edges":["e-f12c666736870d38"]},{"id":158471,"position":6,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 5]","output_interval":"(-inf, 0]","statement":"an input root r <= 5 gives the root (r - a)/d <= 0; a larger root gives a positive root"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 5] to roots in (-inf, 0]: an input root r <= 5 gives the root (r - a)/d <= 0; a larger root gives a positive root","region_exact":null,"edges":["e-f12c666736870d38"]},{"id":158472,"position":7,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-0ab1d5f59b4d29a0"]},{"id":158473,"position":8,"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-0ab1d5f59b4d29a0"]}],"region_exact":{},"incoming_edges":[{"id":"e-8f220a12633250f0","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-7b91a7a037fb3513","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"ed7f972484011d4644ff74144adc1a1f6d2b2094b7c5c4a0a09e45f8d83d8722","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-3dcaebc50e901d48","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-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]"}]},{"id":"e-f12c666736870d38","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-7b91a7a037fb3513","premise_count":0,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"5a398e1fd864a78aaea8f9730991b0b04e481f225711b0755574c4602e8981e7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},{"claim":"basis_map_preserver","input_interval":"(-inf, 5]","output_interval":"(-inf, 0]","statement":"an input root r <= 5 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[]},{"id":"e-868f364735375874","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-7b91a7a037fb3513","premise_count":0,"theorem":"polytool parent provers, route periodic_product: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"5fec6730357960ee6ba359f8542cb0d60f85e1e3020e2a50b6ccbed793d19914","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},{"id":"e-0ab1d5f59b4d29a0","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-7b91a7a037fb3513","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"a8bd35961436a0581c72ccd9090625e89d460a69857fb54147622befb0b1cc29","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3dcaebc50e901d48","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-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]"}]},{"id":"e-f2f89dae9298aff8","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-7b91a7a037fb3513","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"83cc97b64084d3229005a0ac4d53701e56c07d821542db5ac0e3ca3b3ba1fc1f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-037f2552a98ec0cc","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-ff3307aeb79ab12b","conclusion_name":"basis_map_A013609 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j)."},{"id":"e-0e13ca82ccc1a33c","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-408f7506d6594551","conclusion_name":"basis_map_A013611 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j)."},{"id":"e-2d1052dd37eda11b","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-7f05fd9e9c6948d8","conclusion_name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j."},{"id":"e-9e3373425bf2889e","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-384f44bb897362c3","conclusion_name":"basis_map_A008277 of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j)."},{"id":"e-efe1634c04c9236a","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-22693abd431a71f2","conclusion_name":"Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)."},{"id":"e-c4ff17d209cd99f9","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-fb98746e3ace60b9","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j)."},{"id":"e-c4ff17d209cd99f9","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-fb98746e3ace60b9","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j)."}],"used_by_total":7,"links":[{"id":"l-10eff73808e8322c","kind":"fingerprint_related","node_a_id":"n-7b91a7a037fb3513","node_b_id":"n-804e872f93921cf0","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-804e872f93921cf0","other":{"id":"n-804e872f93921cf0","name":"basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","3"],["225","90","9"]],"card_rows":[["1"],["15","3"],["225","90","9"],["3375","2025","405","27"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610"],"edges":["e-011a72a323f5e83e"],"text":"A147716 \u2192 basis A013610"},"headline":"A147716 \u2192 basis A013610","chain":null}},{"id":"l-5ae2a5ff77bc44aa","kind":"fingerprint_related","node_a_id":"n-bbfc5a54ff13a785","node_b_id":"n-7b91a7a037fb3513","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-bbfc5a54ff13a785","other":{"id":"n-bbfc5a54ff13a785","name":"Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","10","25"],["1","15","75","125"],["1","20","150","500","625"]],"card_rows":[["1","10","25"],["1","15","75","125"],["1","20","150","500","625"],["1","25","250","1250","3125","3125"]],"row_sums_preview":["36","216","1296","7776","46656","279936","1679616","10077696"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":[],"edges":[],"text":"A013612"},"headline":"Triangle of coefficients in expansion of (1+5x)^n.","chain":null}},{"id":"l-75830e9c51ec5be5","kind":"fingerprint_related","node_a_id":"n-7b91a7a037fb3513","node_b_id":"n-8447c9ce68adee7a","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-8447c9ce68adee7a","other":{"id":"n-8447c9ce68adee7a","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["225","90","9"],["3375","2025","405","27"],["50625","40500","12150","1620","81"]],"card_rows":[["225","90","9"],["3375","2025","405","27"],["50625","40500","12150","1620","81"],["759375","759375","303750","60750","6075","243"]],"row_sums_preview":["324","5832","104976","1889568","34012224","612220032","11019960576","198359290368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A007318"],"edges":["e-fe430a680aa07591"],"text":"A038329 \u2192 basis A007318"},"headline":"A038329 \u2192 basis A007318","chain":null}},{"id":"l-84fbd94a77c724ad","kind":"fingerprint_related","node_a_id":"n-7b91a7a037fb3513","node_b_id":"n-6323575beb222864","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-6323575beb222864","other":{"id":"n-6323575beb222864","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","2"],["100","40","4"]],"card_rows":[["1"],["10","2"],["100","40","4"],["1000","600","120","8"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609"],"edges":["e-88ca34e9877f4f47"],"text":"A038291 \u2192 basis A013609"},"headline":"A038291 \u2192 basis A013609","chain":null}},{"id":"l-c78a3598bca2332b","kind":"fingerprint_related","node_a_id":"n-7b91a7a037fb3513","node_b_id":"n-b83665b9a9abd39d","fingerprint":"f6-a8e18dd2f0709b33","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8e18dd2f0709b33"},"other_id":"n-b83665b9a9abd39d","other":{"id":"n-b83665b9a9abd39d","name":"mobius_x_over_x_plus_one of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-a8e18dd2f0709b33","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","5"],["1","10","25"]],"card_rows":[["1"],["1","5"],["1","10","25"],["1","15","75","125"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["x \u21a6 x/(x+1)"],"edges":["e-949a223514c284da"],"text":"A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-0d1b2027edbbedb1","kind":"oeis_match","node_a_id":"n-7b91a7a037fb3513","node_b_id":null,"fingerprint":null,"oeis_id":"A013612","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A013612","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-739462a97d6c9d17","kind":"oeis_match","node_a_id":"n-7b91a7a037fb3513","node_b_id":null,"fingerprint":null,"oeis_id":"A038243","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A038243","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-bb48163ffbc38b23","kind":"oeis_match","node_a_id":"n-7b91a7a037fb3513","node_b_id":null,"fingerprint":null,"oeis_id":"A000400","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000400","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A038243","relation":"identification"},{"oeis_id":"A013612","relation":"match_reversed_triangle"},{"oeis_id":"A000400","relation":"match_row_sums"},{"oeis_id":"A038243","relation":"match_triangle"}],"bfiles":[{"node_id":"n-7b91a7a037fb3513","oeis_id":"A000400","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":16479},{"node_id":"n-7b91a7a037fb3513","oeis_id":"A013612","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":20569},{"node_id":"n-7b91a7a037fb3513","oeis_id":"A038243","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":20569}],"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:       1;\n  1:       5,       1;\n  2:      25,      10,       1;\n  3:     125,      75,      15,       1;\n  4:     625,     500,     150,      20,      1;\n  5:    3125,    3125,    1250,     250,     25,     1;\n  6:   15625,   18750,    9375,    2500,    375,    30,     1;\n  7:   78125,  109375,   65625,   21875,   4375,   525,    35,   1;\n  8:  390625,  625000,  437500,  175000,  43750,  7000,   700,  40,  1;\n  9: 1953125, 3515625, 2812500, 1312500, 393750, 78750, 10500, 900, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(5 + x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5 + x)*v[i - 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}},{"oeis_id":"A013612","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  5;\n  2: 1, 10,  25;\n  3: 1, 15,  75,   125;\n  4: 1, 20, 150,   500,   625;\n  5: 1, 25, 250,  1250,  3125,   3125;\n  6: 1, 30, 375,  2500,  9375,  18750,   15625;\n  7: 1, 35, 525,  4375, 21875,  65625,  109375,   78125;\n  8: 1, 40, 700,  7000, 43750, 175000,  437500,  625000,  390625;\n  9: 1, 45, 900, 10500, 78750, 393750, 1312500, 2812500, 3515625, 1953125;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (5+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(5 + x)*P[n - 1, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5 + x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A038243","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:       1;\n  1:       5,       1;\n  2:      25,      10,       1;\n  3:     125,      75,      15,       1;\n  4:     625,     500,     150,      20,      1;\n  5:    3125,    3125,    1250,     250,     25,     1;\n  6:   15625,   18750,    9375,    2500,    375,    30,     1;\n  7:   78125,  109375,   65625,   21875,   4375,   525,    35,   1;\n  8:  390625,  625000,  437500,  175000,  43750,  7000,   700,  40,  1;\n  9: 1953125, 3515625, 2812500, 1312500, 393750, 78750, 10500, 900, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(5 + x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5 + x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-7b91a7a037fb3513","steps":[],"edges":[],"text":"A038243"},"headline":"A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","chain":null}